I 


UC-NRLF 


and  Philosophy. 


THE 


GEOMETRICAL  LECTURES 


ISAAC  BARROW 


J.     M.    CHILD 


J 


THE 

GEOMETRICAL    LECTURES 

OF 

ISAAC   BARROW 


The  Open  Court  Series  of  Classics  of  Science  and 
Thilosophy,  U^o.  3 

THE 

EOMETRICAL  LECTURES 

OF 

ISAAC   BARROW 


TRANSLATED,  WITH  NOTES  AND  PROOFS,  AND  A 

DISCUSSION  ON  THE  ADVANCE  MADE  THEREIN 

ON  THE  WORK  OF   HIS   PREDECESSORS  IN   THE 

INFINITESIMAL  CALCULUS 


BY 


J.  M.  CHILD 

B.A.  (CANTAB.),  B.Sc.  (LOND.) 


CHICAGO   AND    LONDON 
THE  OPEN  COURT  PUBLISHING  COMPANY 

1916 


Copyright  in  Great  Britain  tinder  the  Act  of  1911 


LECTI ONES 

Geometric^ ; 

In  quibus  (prarfertim) 

GEN.ERALJA    Cwcvarum   'Linear urn  SYMPTOUATA 
D  E  C  L  A  T    A         T  U  T. 


Au&ore  ISAACC>BARROW  Collegii 

SS.  Triyitattf  in  Acad.  Cantab.  SociD,  &  Societalis  tic- 
?t<c  Sodale. 


Oi  <fv 

OJT«    /S^e/Wj  j     tt^  t»    rare*  Tnuii          ^upr«on»j'7tt/ 
AAo   ai'^tAHj,^^,  o^wc  «c-;«  TO 
eifiv.   Plato  de  Repub.  VII. 


L  ON  D  IN  I, 

Typis  Gultelmi  Godbid ,  &  proftant    venales  apud 

Joh.tnntm  T)*nm3re,  &  OttavLmam  F alley n  Juniorem. 

UW.  D     .  L  r  A". 


Note  the  absence  of  the  usual  words  "  Habitae  Cantabrigise,"  which  on 
the  title-pages  of  his  other  works  indicate  that  the  latter  were  delivered  as 
Lucasian  Lectures. — J.  M.  C. 


382569 


PREFACE 

ISAAC  BARROW  was  the  first  inventor  of  the  Infinitesimal 
Calculus  ;  Newton  got  the  main  idea  of  it  from  Barrow  by 
personal  communication;  and  Leibniz  also  was  in  some 
measure  indebted  to  Barrow's  work,  obtaining  confirmation 
of  his  own  original  ideas,  and  suggestions  for  their  further 
development,  from  the  copy  of  Bar  row*  s  book  that  he  purchased 
in  1673. 

The  above  is  the  ultimate  conclusion  that  I  have  arrived 
at,  as  the  result  of  six  months'  close  study  of  a  single  book, 
my  first  essay  in  historical  research.  By  the  "Infinitesimal 
Calculus,"  1  intend  "a  complete  set  of  standard  forms  for 
both  the  differential  and  integral  sections  of  the  subject, 
together  with  rules  for  their  combination,  such  as  for  a 
product,  a  quotient,  or  a  power  of  a  function ;  and  also  a 
recognition  and  demonstration  of  the  fact  that  differentiation 
and  integration  are  inverse  operations." 

The  case  of  Newton  is  to  my  mind  clear  enough.  Barrow 
was  familiar  with  the  paraboliforms,  and  tangents  and  areas 
connected  with  them,  in  from  1655  to  1660  at  the  very 
latest;  hence  he  could  at  this  time  differentiate  and  inte- 
grate by  his  own  method  any  rational  positive  power  of  a 
variable,  and  thus  also  a  sum  of  such  powers.  He  further 
developed  it  in  the  years  1662-3-4,  and  in  the  latter  year 
probably  had  it  fairly  complete.  In  this  year  he  com- 
municated to  Newton  the  great  secret  of  his  geometrical 
constructions,  as  far  as  it  is  humanly  possible  to  judge  from 
a  collection  of  tiny  scraps  of  circumstantial  evidence ;  and 
it  was  probably  this  that  set  Newton  to  work  on  an  attempt 
to  express  everything  as  a  sum  of  powers  of  the  variable. 
During  the  next  year  Newton  began  to  "reflect  on  his 
method  of  fluxions,"  and  actually  did  produce  his  Analysis 
per  &quationes.  This,  though  composed  in  1666,  was  not 
published  until  1711. 


viii   BARROW'S  GEOMETRICAL  LECTURES 

The  case  of  Leibniz  wants  more  argument  that  I  am  in  a 
position  at  present  to  give,  nor  is  this  the  place  to  give  it.  I 
hope  to  be  able  to  submit  this  in  another  place  at  some  future 
time.  The  striking  points  to  my  mind  are  that  Leibniz 
bought  a  copy  of  Barrow's  work  in  1673,  and  was  able  "to 
communicate  a  candid  account  of  his  calculus  to  Newton  " 
in  1677.  In  this  connection,  in  the  face  of  Leibniz'  per- 
sistent denial  that  he  received  any  assistance  whatever  from 
Barrow's  book,  we  must  bear  well  in  mind  Leibniz'  twofold 
idea  of  the  "calculus":— 

(i)  the  freeing  of  the  matter  from  geometry, 
(ii)  the  adoption  of  a  convenient  notation. 
Hence,  be  his  denial  a  mere  quibble  or  a  candid  statement 
without  any  thought  of  the  idea  of  what  the  "  calculus " 
really  is,  it  is  perfectly  certain  that  on  these  two  points  at 
any  rate  he  derived  not  the  slightest  assistance  from 
Barrow's  work ;  for  the  first  of  them  would  be  dead  against 
Barrow's  practice  and  instinct,  and  of  the  second  Barrow 
had  no  knowledge  whatever.  These  points  have  made  the 
calculus  the  powerful  instrument  that  it  is,  and  for  this  the 
world  has  to  thank  Leibniz;  but  their  inception  does  not 
mean  the  invention  of  the  infinitesimal  calculus.  This,  the 
epitome  of  the  work  of  his  predecessors,  and  its  completion 
by  his  own  discoveries  until  it  formed  a  perfected  method 
of  dealing  with  the  problems  of  tangents  and  areas  for 
any  curve  in  general,  i.e.  in  modern  phraseology,  the 
differentiation  and  integration  of  any  function  whatever 
(such  as  were  known  in  Barrow's  time),  must  be  ascribed 
to  Barrow. 

Lest  the  matter  that  follows  may  be  considered  rambling, 
and  marred  by  repetitions  and  other  defects,  I  give  first 
some  account  of  the  circumstances  that  gave  rise  to  this 
volume.  First  of  all,  I  was  asked  by  Mr  P.  E.  B.  Jourdain 
to  write  a  short  account  of  Barrow  for  the  Monist ;  the 
request  being  accompanied  by  a  first  edition  copy  of 
Barrow's  Lectio?ies  Opticce.  et  Geometricce.  At  this  time,  I 
do  not  mind  confessing,  my  only  knowledge  of  Barrow's 
claim  to  fame  was  that  he  had  been  "Newton's  tutor":  a 
notoriety  as  unenviable  as  being  known  as  "  Mrs  So-and-So's 
husband."  For  this  article  I  read,  as  if  for  a  review,  the 
book  that  had  been  sent  to  me.  My  attention  was  arrested 


PREFACE  ix 

by  a  theorem  in  which  Barrow  had  rectified  the  cycloid,  which 
I  happened  to  know  has  usually  been  ascribed  to  Sir  C.  Wren. 
My  interest  thus  aroused  impelled  me  to  make  a  laborious 
(for  I  am  no  classical  scholar)  translation  of  the  whole  of 
the  geometrical  lectures,  to  see  what  else  I  could  find.  The 
conclusions  I  arrived  at  were  sent  to  the  Monist  for  publica- 
tion ;  but  those  who  will  read  the  article  and  this  volume 
will  find  that  in  the  article  I  had  by  no  means  reached  the 
stage  represented  by  this  volume.  Later,  as  I  began  to  still 
further  appreciate  what  these  lectures  really  meant,  I  con- 
ceived the  idea  of  publishing  a  full  translation  of  the  lectures 
together  with  a  summary  of  the  work  of  Barrow's  more 
immediate  predecessors,  written  in  the  same  way  from  a 
personal  translation  of  the*  originals,  or  at  least  of  all  those 
that  I  could  obtain.  On  applying  to  the  University  Press, 
Cambridge,  through  my  friend,  the  Rev.  J.  B.  Lock,  I  was 
referred  by  Professor  Hobson  to  the  recent  work  of  Professor 
Zeuthen.  On  communicating  with  Mr  Jourdain,  I  was 
invited  to  elaborate  my  article  for  the  Monist  into  a 
2oo-page  volume  for  the  Open  Court  Series  of  Classics. 

I  can  lay  no  claim  to  any  great  perspicacity  in  this  dis- 
covery of  mine,  if  I  may  call  it  so ;  all  that  follows  is  due 
rather  to  the  lack  of  it,  and  to  the  lucky  accident  that  made 
me  (when  I  could  not  follow  the  demonstration)  turn  one 
of  Barrow's  theorems  into  algebraical  geometry.  What  I 
found  induced  me  to  treat  a  number  of  the  theorems  in  the 
same  way.  As  a  result  I  came  to  the  conclusion  that 
Barrow  had  got  the  calculus;  but  I  queried  even  then 
whether  Barrow  himself  recognized  the  fact.  Only  on  com- 
pleting my  annotation  of  the  last  chapter  of  this  volume, 
Lect.  XII,  App.  Ill,  did  I  come  to  the  conclusion  that  is 
given  as  the  opening  sentence  of  this  Preface ;  for  I  then 
found  that  a  batch  of  theorems  (which  I  had  on  first  reading 
noted  as  very  interesting,  but  not  of  much  service),  on  careful 
revision,  turned  out  to  be  the  few  missing  standard  forms, 
necessary  for  completing  the  set  for  integration  ;  and  that  one 
of  his  problems  was  a  practical  rule  for  finding  the  area 
under  any  curve,  such  as  would  not  yield  to  the  theoretical 
rules  he  had  given,  under  the  guise  of  an  "inverse-tangent" 
problem. 

The  reader  will  then  understand  that  the  conclusion  is 


x      BARROW'S  GEOMETRICAL  LECTURES 

the  effect  of  a  gradual  accumulation  of  evidence  (much  as 
a  detective  picks  up  clues)  on  a  mind  previously  blank  as 
regards  this  matter,  and  therefore  perfectly  unbiased.  This 
he  will  see  reflected  in  the  gradual  transformation  from 
tentative  and  imaginative  suggestions  in  the  Introduction 
to  direct  statements  in  the  notes,  which  are  inset  in  the 
text  of  the  latter  part  of  the  translation.  I  have  purposely 
refrained  from  altering  the  Introduction,  which  preserves  the 
form  of  my  article  in  the  Monist,  to  accord  with  my  final 
ideas,  because  I  feel  that  with  the  gradual  development 
thus  indicated  I  shall  have  a  greater  chance  of  carrying  my 
readers  with  me  to  my  own  ultimate  conclusion. 

The  order  of  writing  has  been  (after  the  first  full  trans- 
lation had  been  made): — Introduction,  Sections  I  to  VIII, 
excepting  III ;  then  the  text  with  notes;  then  Sections  III 
and  IX  of  the  Introduction ;  and  lastly  some  slight  altera- 
tions in  the  whole  and  Section  X. 

In  Section  I,  I  have  given  a  wholly  inadequate  account 
of  the  work  of  Barrow's  immediate  predecessors ;  but  I  felt 
that  this  could  be  enlarged  at  any  reader's  pleasure,  by 
reference  to  the  standard  historical  authorities ;  and  that  it 
was  hardly  any  of  my  business,  so  long  as  I  slightly  expanded 
rriy  Monist  article  to  a  sufficiency  for  the  purpose  of  showing 
that  the  time  was  now  ripe  for  the  work  of  Barrow,  Newton, 
and  Leibniz.  This,  and  the  next  section,  have  both  been 
taken  from  the  pages  of  the  Encyclopedia  Britannica  ( Times 
edition). 

The  remainder  of  my  argument  simply  expresses  my  own, 
as  I  have  said,  gradually  formed  opinion.  I  have  purposely 
refrained  from  consulting  any  authorities  other  than  the 
work  cited  above,  the  Bibliotheca  Britannica  (for  the  dates  in 
Section  III),  and  the  Dictionary  of  National  Biography  (for 
Canon  Overton's  life  of  Barrow) ;  but  I  must  acknowledge 
the  service  rendered  me  by  the  dates  and  notes  in  Sotheran's 
Price  Current  of  Literature.  The  translation  too  is  entirely 
my  own — without  any  help  from  the  translation  by  Stone 
or  other  assistance — from  a  first  edition  of  Barrow's  work 
dated  1670. 

As  regards  the  text,  with  my  translation  beside  me,  I 
have  to  all  intents  rewritten  Barrow's  book;  although 
throughout  I  have  adhered  fairly  closely  to  Barrow's  own 


PREFACE  xi 

words.  I  have  only  retained  those  parts  which  seemed  to 
me  to  be  absolutely  essential  for  the  purpose  in  hand. 
Thus  the  reader  will  find  the  first  few  chapters  very  much 
abbreviated,  not  only  in  the  matter  of  abridgment,  but  also 
in  respect  of  proofs  omitted,  explanations  cut  down,  and 
figures  left  out,  whenever  this  was  possible  without  breaking 
the  continuity.  This  was  necessary  in  order  that  room 
might  be  found  for  the  critical  notes  on  the  theorems,  the 
inclusion  of  proofs  omitted  by  Barrow,  which  when  given 
in  Barrow's  style,  and  afterwards  translated  into  analysis, 
had  an  important  bearing  on  the  point  as  to  how  he  found 
out  the  more  difficult  of  his  constructions;  and  lastly  for 
deductions  therefrom  that  point  steadily,  one  after  the 
other,  to  the  fact  that  Barrow  was  writing  a  calculus  and  : 
knew  that  he  was  inventing  a  great  thing.  I  can  make  no 
claim  to  any  classical  attainments,  but  I  hope  the  transla- 
tion will  be  found  correct  in  almost  every  particular.  In 
the  wording  I  have  adhered  to  the  order  in  which  the 
original  runs,  because  thereby  the  old-time  flavour  is  not 
lost ;  the  most  I  have  done  is  to  alter  a  passage  from  the 
active  to  the  passive  or  vice  versa,  and  occasionally  to 
change  the  punctuation. 

I  have  used  three  distinct  kinds  of  type :  the  most  widely 
spaced  type  has  been  used  for  Barrow's  own  words ;  only 
very  occasionally  have  I  inserted  anything  of  my  own  in 
this,  and  then  it  will  be  found  enclosed  in  heavy  square 
brackets,  that  the  reader  will  have  no  chance  of  confusing 
my  explanations  with  the  text ;  the  whole  of  the  Introduc- 
tion, including  Barrow's  Prefaces,  is  in  the  closer  type ; 
this  type  is  also  used  for  my  critical  notes,  which  are 
generally  given  at  the  end  of  a  lecture,  bui  also  sometimes 
occur  at  the  end  of  other  natural  divisions  of  the  work, 
when  it  was  thought  inadvisable  to  put  off  the  explanation 
until  the  end  of  the  lecture.  It  must  be  borne  in  mind 
that  Barrow  makes  use  of  parentheses  very  frequently,  so 
that  the  reader  must  understand  that  only  remarks  in  heavy 
square  brackets  are  mine,  those  in  ordinary  round  brackets 
are  Barrow's.  The  small  type  is  used  for  footnotes  only. 
In  the  notes  I  have  not  hesitated  to  use  the  Leibniz 
notation,  because  it  will  probably  convey  my  meaning 
better ;  but  there  was  really  no  absolute  necessity  for  this, 


xii   BARROW'S  GEOMETRICAL  LECTURES 

Barrow's  a  and  e,  or  its  modern  equivalent,  h  and  /£,  would 
have  done  quite  as  well. 

I  cannot  close  this  Preface  without  an  acknowledgment 
of  my  great  indebtedness  to  Mr  Jourdain  for  frequent 
advice  and  help;  I  have  had  an  unlimited  call  on  his  wide 
reading  and  great  historical  knowledge ;  in  fact,  as  Barrow 
says  of  Collins,  I  am  hardly  doing  him  justice  in  calling  him 
my  "  Mersenne."  All  the  same,  I  accept  full  responsibility 
for  any  opinions  that  may  seem  to  be  heretical  or  otherwise 
out  of  order.  My  thanks  are  also  due  to  Mr  Abbott,  of 
Jesus  College,  Cambridge,  for  his  kind  assistance  in  looking 
up  references  that  were  inaccessible  to  me. 

J.  M.  CHILD. 
DEKKY,  ENGLAND, 
Xnias,   1915. 

P.S. — Since  this  volume  has  been  ready  for  press,  I  have 
consulted  several  .authorities,  and,  through  the  kindness  of 
Mr  Walter  Stott,  I  have  had  the  opportunity  of  reading 
Stone's  translation.  The  result  I«  have  set  in  an  appendix 
at  the  end  of  the  book.  The  reader  will  also  find  there  a 
solution,  by  Barrow's  methods,  of  a  test  question  suggested 
by  Mr  Jourdain  ;  after  examining  this  I  doubt  whether  any 
reader  will  have  room  for  doubt  concerning  the  correctness 
of  my  main  conclusion.  I  have  also  given  two  specimen 
pages  of  Barrow's  text  and  a  specimen  of  his  folding  plates 
of  diagrams.  Also,  I  have  given  an  example  of  Barrow's 
graphical  integration  of  a  function ;  for  this  I  have  chosen 
a  function  which  he  could  not  have  integrated  theoretically, 
namely,  1/^(1  ~^4),  between  the  limits  o  and  x ;  when  the 
upper  limit  has  its  maximum  value,  i,  it  is  well  known  that 
the  integral  can  be  expressed  in  Gamma  functions ;  this 
was  used  as  a  check  on  the  accuracy  of  the  method. 

J.  M.  C. 


TABLE    OF   CONTENTS 

PAGE 

INTRODUCTION — 

The  work  of  Barrow's  great  predecessors         ....  I 

Life  of  Barrow,  and  his  connection  with  Newton    .  6 

The  works  of  Barrow      ......  & 

Estimate  of  Barrow's  genius  ....  9 

The  sources  of  Barrow's  ideas         ....  .12 

Mutual  influence  of  Newton  and  Barrow          ....          16 

Description  of  the  book  from  which  the  translation  has  been 

made  ..........         20 

The  prefaces -25 

How  Barrow  made  his  constructions       ...  .28 

Analytical  equivalents  of  Barrow's  chief  theorems   ...          30 

TRANSLATION — 

LECTURE  I. —Generation  of  magnitudes.  Modes  of  motion  and 
the  quantity  of  the  motive  force.  Time  as  the  independent 
variable.  Time,  as  an  aggregation  of  instants,  compared 
with  a  line,  as  an  aggregation  of  points  35 

LECTURE  II. — Generation  of  magnitudes  by  "local  move- 
ments." The  simple  motions  of  translation  and  rotation  .  42 

LECTURE  III. — Composite  and  concurrent  motions.  Com- 
position of  rectilinear  and  parallel  motions  .  .  .  47 

LECTURE  IV. — Properties  of  curves  arising  from  composition 
of  motions.  The  gradient  of  the  tangent.  Generalization 
of  a  problem  of  Galileo.  Case  of  infinite  velocity  .  .  53 

LECTURE  V. — Further  properties  of  curves.   Tangents.    Curves 

like  the  cycloid.     Normals.     Maximum  and  minimum  lines          60 

LECTURE  VI. — Lemmas  ;  determination  of  certain  curves  con- 
structed according  to  given  conditions  ;  mostly  hyperbolas  .  69 

LECTURE  VII.  — Similar  or  analogous  curves.  Exponents 
or  Indices.  Arithmetical  and  Geometrical  Progressions. 
Theorem  analogous  to  the  approximation  to  the  Binomial 
Theorem  for  a  Fractional  Index.  Asymptotes  .  ,  77 


xiv  BARROW'S  GEOMETRICAL  LECTURES 

LECTURE  VIII. — Construction  of  tangents  by  means  of  auxiliary 
curves  of  which  the  tangents  are  known.  Differentiation 
of  a  sum  or  a  difference.  Analytical  equivalents  .  .  90 

LECTURE  IX. — Tangents  to  curves  formed  by  arithmetical  and 
geometrical  means.  Paraboliforms.  Curves  of  hyperbolic 
and  elliptic  form.  Differentiation  of  a  fractional  power, 
products  and  quotients  .......  101 

LECTURE  X. — Rigorous  determination  of  dsjdx.  Differentia- 
tion as  the  inverse  of  integration.  Explanation  of  the 
"Differential  Triangle"  method;  with  examples.  Differ- 
entiation of  a  trigonometrical  function  .  .  .  .  113 

LECTURE  XI. — Change  of  the  independent  variable  in  inte- 
gration. Integration  the  inverse  of  differentiation.  Differ- 
entiation of  a  quotient.  Area  and  centre  of  gravity  of  a 
paraboliform.  Limits  for  the  arc  of  a  circle  and  a  hyperbola. 
Estimation  of  TT  .  .  .  .  .  .  .  .  .125 

LECTURE  XII. — General  theorems  on  rectification.  Standard 
forms  for  integration  of  circular  functions  by  reduction  to 
the  quadrature  of  the  hyperbola.  Method  of  circumscribed 
and  inscribed  figures.  Measurement  of  conical  surfaces. 
Quadrature  of  the  hyperbola.  Differentiation  and  Integra- 
tion of  a  Logarithm  and  an  Exponential.  Further  standard 
forms  .........  155 

LECTURE  XIII. —These  theorems  have  not  been  inserted        .        196 

POSTSCRIPT — 

Extracts  from  Standard  Authorities     .          .          .          .         .198 

APPENDIX — 

I.   Solution  of  a  test  question  by  Barrow's  method  .  .        207 

II.   Graphical  integration  by  Barrow's  method       .  .  .211 

III.   Reduced  facsimiles  of  Barrow's  pages  and  figures  .  .        212 

INDEX 216 


INTRODUCTION 


THE  WORK  OF  BARROW'S  GREAT 
PREDECESSORS 

THE  beginnings  of  the  Infinitesimal  Calculus,  in  its  two 
main  divisions,  arose  from  determinations  of  areas  and 
volumes,  and  the  finding  of  tangents  to  plane  curves.  The 
ancients  attacked  the  problems  in  a  strictly  geometrical 
manner,  making  use  of  the  "  method  of  exhaustions."  In 
modern  phraseology,  they  found  "upper  and  lower  limits," 
as  closely  equal  as  possible,  between  which  the  quantity 
to  be  determined  must  lie ;  or,  more  strictly  perhaps,  they 
showed  that,  if  the  quantity  could  be  approached  from  two 
"  sides,"  on  the  one  side  it  was  always  greater  than  a  certain 
thing,  and  on  the  other  it  was  always  less ;  hence  it  must  be 
finally  equal  to  this  thing.  This  was  the  method  by  means 
of  which  Archimedes  proved  most  of  his  discoveries.  But 
there  seems  to  have  been  some  distrust  of  the  method,  for 
we  find  in  many  cases  that  the  discoveries  are  proved  by  a 
reductio  ad  absurdum,  such  as  one  is  familiar  with  in  Euclid. 
To  Apollonius  we  are  indebted  for  a  great  many  of  the  pro- 
perties, and  to  Archimedes  for  the  measurement,  of  the  conic 
sections  and  the  solids  formed  from  them  by  their  rotation 
about  an  axis. 

The  first  great  advance,  after  the  ancients,  came  in  the 
beginning  of  the  seventeenth  century.  Galileo  (1564-1642) 
would  appear  to  have  led  the  way,  by  the  introduction  of 
the  theory  of  composition  of  motions  into  mechanics ;  *  he 
also  was  one  of  the  first  to  use  infinitesimals  in  geometry, 
and  from  the  fact  that  he  uses  what  is  equivalent  to  "virtual 
velocities"  it  is  to  be  inferred  that  the  idea  of  time  as  the, 
independent  variable  is  due  to  him.  Kepler  (1571-1630) 
was  the  first  to  introduce  the  idea  of  infinity  into  geometry 

*  See  Mach's  Science  of  Mechanics  for  fuller  details. 

I 


2      SARROV/'S  GEOMETRICAL  LECTURES 

ar/d  to  net?  that  the  increment  of  a  variable  was  evanescent 
for  values  of  the  variable  in  the  immediate  neighbourhood 
of  a  maximum  or  minimum;  in  1613,  an  abundant  vintage 
drew  his  attention  to  the  defective  methods  in  use  for 
estimating  the  cubical  contents  of  vessels,  and  his  essay 
on  the  subject  (Nova  Stereometria  Doliorum]  entitles  him 
to  rank  amongst  those  who  made  the  discovery  of  the  in- 
-finitesimal  calculus  possible.  In  1635,  Cavalieri  published 
a  theory  of  "indivisibles,"  in  which  he  considered  a  line  as 
made  up  of  an  infinite  number  of  points,  a  superficies  as 
composed  of  a  succession  of  lines,  and  a  solid  as  a  succession 
of  superficies;  thus  laying  the  foundation  for  the  "aggre- 
gations "  of  Barrow.  Roberval  seems  to  have  been  the  first, 
or  at  the  least  an  independent,  inventor  of  the  method  ;  but 
he  lost  credit  for  it,  because  he  did  not  publish  it,  preferring 
to  keep  the  method  to  himself  for  his  own  use ;  this  seems 
to  have  been  quite  a  usual  thing  amongst  learned  men  of 
that  time,  due  perhaps  to  a  certain  professional  jealousy. 
The  method  was  severely  criticized  by  contemporaries, 
especially  by  Guldin,  but  Pascal  (1623-1662)  showed  that 
the  method  of  indivisibles  was  as  rigorous  as  the  method 
of  exhaustions,  in  fact  that  they  were  practically  identical. 
In  all  probability  the  progress  of  mathematical  thought  is 
much  indebted  to  this  defence  by  Pascal.  Since  this  method 
is  exactly  analogous  to  the  ordinary  method  of  integration, 
Cavalieri  and  Roberval  have  more  than  a  little  claim  to  be 
regarded  as  the  inventors  of  at  least  the  one  branch  of  the 
calculus  ;  if  it  were  not  for  the  fact  that  they  only  applied  it 
to  special  cases,  and  seem  to  have  been  unable  to  generalize 
it  owing  to  cumbrous  algebraical  notation,  or  to  have  failed 
to  perceive  the  inner  meaning  of  the  method  when  concealed 
under  a  geometrical  form.  Pascal  himself  applied  the 
method  with  great  success,  but  also  to  special  cases  only ; 
such  as  his  work  on  the  cycloid.  The  next  step  was  of  a 
more  analytical  nature;  by  the  method  of  indivisibles, 
Wallis  (1616-1703)  reduced  the  determination  of  many 
areas  and  volumes  to  the  calculation  of  the  value  of  the 
series  (om  +  im  +  2m  +  .  .  .  nm)j(n  +  i)nm,  i.e.  the  ratio  of  the 
mean  of  all  the  terms  to  the  last  term,  for  integral  values  of  n  ; 
and  later  he  extended  his  method,  by  a  theory  of  interpola- 
tion, to  fractional  values  of  n.  Thus  the  idea  of  the  Integral 


INTRODUCTION  3 

Calculus  was  in  a  fairly  advanced  stage  in  the  days  immedi- 
ately antecedent  to  Barrow. 

What  Cavalieri  and  Roberval  did  for  the  integral  calculus, 
Descartes  (1596-1650)  accomplished  for  the  differential 
branch  by  his  work  on  the  application  of  algebra  to  geometry. 
Cartesian  coordinates  made  possible  the  extension  of  in- 
vestigations on  the  drawing  of  tangents  to  special  curves  to 
the  more  general  problem  for  curves  of  any  kind.  To  this 
must  be  added  the  fact  that  he  habitually  used  the  index 
notation  •  for  this  had  a  very  great  deal  to  do  with  the 
possibility  of  Newton's  discovery  of  the  general  binomial 
expansion  and  of  many  other  infinite  series.  Descartes 
failed,  however,  to  make  any  very  great  progress  on  his  own 
account  in  the  matter  of  the  drawing  of  tangents,  owing  to 
what  I  cannot  help  but  call  an  unfortunate  choice  of  a 
definition  for  a  tangent  to  a  curve  in  general.  Euclid's 
circle-tangent  definition  being  more  or  less  hopeless  in  the 
general  case,  Descartes  had  the  choice  of  three : — 

(1)  a  secant,  of  which  the  points  of  intersection  with 

the  curve  became  coincident ; 

(2)  a  prolongation  of  an  element  of  the  curve,  which 

was  to  be  considered  as  composed  of  an  infinite 
succession  of  infinitesimal  straight  lines ; 

(3)  the  direction  of  the  resultant  motion,  by  which  the 

curve  might  have  been  described. 

Descartes  chose  the  first ;  I  have  called  this  choice  unfor- 
tunate, because  I  cannot  see  that  it  would  have  been  possible 
for  a  Descartes  to  miss  the  differential  triangle,  and  all  its 
consequences,  if  he  had  chosen  the  second  definition.  His 
choice  leads  him  to  the  following  method  of  drawing  a 
tangent  to  a  curve  in  general.  Describe  a  circle,  whose 
centre  is  on  the  axis  of  x,  to  cut  the  curve ;  keeping  the 
centre  fixed,  diminish  the  radius  until  the  points  of  section 
coincide ;  thus,  by  the  aid  of  the  equation  of  the  curve,  the 
problem  is  reduced  to  finding  the  condition  for  equal  roots 
of  an  equation. 

For  instance,  let  y^^^ax  be  the  equation  to  a  parabola, 
and  (x-p)'2+y2  =  r2  the  equation  of  the  circle.  Then  we 
have  (x-pf  +  ±ax  =  r\  If  this  is  a  perfect  square, 
x=p  —  za \  i.e.  the  subtangent  is  equal  to  2a. 


4      BARROW'S  GEOMETRICAL  LECTURES 

The  method,  however,  is  only  applicable  to  a  small 
number  of  simple  cases,  owing  to  algebraical  difficulties. 
In  the  face  of  this  disability,  it  is  hard  to  conjecture  why 
Descartes  did  not  make  another  choice  of  definition  and  use 
the  second  one  given  above ;  for  in  his  rule  for  the  tangents 
to  roulettes,  he  considers  a  curve  as  the  ultimate  form  of  a 
^polygon.  The  third  definition,  if  not  originally  due  to 
Galileo,  was  a  direct  consequence  of  his  conception  of  the 
composition  of  motions ;  this  definition  was  used  by 
Rpberval  (1602-1675)  and  applied  successfully  to  a  dozen 
or  so  of  the  well-known  curves ;  in  it  we  have  the  germ  of 
the  method  of  "fluxions."  Thus  it  is  seen  that  Roberval 
occupies  an  almost  unique  position,  in  that  he  took  a  great 
part  in  the  work  preparatory  to  the  invention  viboth  branches 
of  the  infinitesimal  calculus ;.  a  fact  that  seems  to  have 
I  escaped  remark.  Fermat  (1590-1663)  adopted  Kepler's 
notion  of  the  increment  of  the  variable  becoming  evanes- 
cent near  a  maximum  or  minimum  value,  and  upon  it 
based  his  method  of  drawing  tangents.  Fermat's  method 
of  finding  the  maximum  or  minimum  value  of  a  function  in- 
volved the  differentiation  of  any  explicit  algebraic  function, 
in  the  form  that  appears  in  any  beginner's  text-book  of  to- 
day (though  Fermat  does  not  seem  to  have  the  "  function  " 
idea) ;  that  is,  the  maximum  or  minimum  values  of  f(x)  are 
the  roots  of  /'(#)  =  °»  wnere  f(x)  is  the  limiting  value  of 
[f(x  +  K)  -f(x]\\h ;  only  Fermat  uses  the  letter  e  or  E  instead 
of  h.  Now,  if  YYY  is  any  curve,  wholly  con- 
cave (or  convex)  to  a  straight  line  AD,  TZYZ 
a  tangent  to  it  at  the  point  Y  whose  ordinate 
is  NY,  and  the  tangent  meets  AD  in  T; 
also,  if  ordinates  NYZ  are  drawn  on  either 
side  of  NY,  cutting  the  curve  in  Y  and  the 
tangent  in  Z;  then  it  is  plain  that  the 
ratio  YN  :  NT  is  a  maximum  (or  a  mini- 
mum) when  Y  is  the  point  of  contact  of 
the  tangent. 

Here  then  we  have  all  the  essentials  for  the  calculus ; 
but  only  for  explicit  integral  algebraic  functions,  needing 
the  binomial  expansion  of  Newton,  or  a  general  method  of 
rationalization  which  did  not  impose  too  great  algebraic 
difficulties,  for  their  further  development;  also,  on  the 


INTRODUCTION  5 

authority  of  Poisson,  Fermat  is  placed  out  of  court,  in  that  he 
also  only  applied  his  method  to  certain  special  cases.  Follow- 
ing the  lead  of  Roberval,  Newton  subsequently  used  the 
third  definition  of  a  tangent,  and  the  idea  of  time  as  th6 
independent  variable,  although  this  was  only  to  insure  that 
one  at  least  of  his  working  variables  should  increase  uni- 
formly. This  uniform  increase  of  the  independent  variable 
would  seem  to  have  been  usual  for  mathematicians  of  th^ 
period  and  to  have  persisted  for  some  time  ;;tfor  later  we  find 
with  Leibniz  and  the  Bernoullis  that  d(dy\dx)  =  (d?yldx*)dx. 
Barrow  also  used  time  as  the  independent  variable  in  order] 
that,  like  Newton,  he  might  insure  that  one  of  his  variables, 
a  moving  point  or  line  or  superficies,  should  proceed  uni-( 
formly;/it  is  to  be  noted,  however,  that  this  is  only  in  the 
lectures1  that  were  added  as  an  afterthought  to  the  strictly 
geometrical  lectures,  and  that  later  this  idea  becomes 
altogether  subsidiary.  Barrow,  however,  chose  his  own 
definition  of  a  tangent,  the  second  of  those  given  above; 
and  to  this  choice  is  due  in  great  measure  his  advance  over 
his  predecessors.  For  his  areas  and  volumes  he  followed 
the  idea  of  Cavalieri  and  Roberval. 

Thus  we  see  that  in  the  time  of  Barrow,  Newton,  and 
Leibniz  the  ground  had  been  surveyed,  and  in  many  direc- 
tions levelled;  all  the  material  was  at  hand,  and  it  only 
wanted  the  master  mind  to  "finish  the  job."  This  was 
possible  in  two  directions,  by  geometry  or  by  analysis ; 
each  method  wanted  a  master  mind  of  a  totally  different 
type,  and  the  men  were  forthcoming.  For  geometry, 
Barrow:  for  analysis,  Newton,  and  Leibniz  with  his  in- 
spiration in  the  matter  of  the  application  of  the  simple  and 
convenient  notation  of  his  calculus  of  finite  differences  to 
infinitesimals  and  to  geometry.  With  all  due  honour  to 
these  three  mathematical  giants,  however,  I  venture  to  assert 
that  their  discoveries  would  have  been  well-nigh  impossible 
to  them  if  they  had  lived  a  hundred  years  earlier;  with  the 
possible  exception  of  Barrow,  who,  being  a  geometer,  was 
more  dependent  on  the  ancients  and  less  on  the  moderns 
of  his  time  than  were  the  two  analysts,  they  would  have 
been  sadly  hampered  but  for  the  preliminary  work  of 
Descartes  and  the  others  I  have  mentioned  (and  some  I 
have  not — such  as  Oughtred),  but  especially  Descartes. 


6      BARROWS  GEOMETRICAL  LECTURES 

II 

LIFE   OF   BARROW,   AND   HIS   CONNECTION 
WITH   NEWTON 

Isaac  Barrow  was  born  in  1630,  the  son  of  a  linen-draper 
in  London.  He  was  first  sent  to  the  Charterhouse  School, 
where  inattention  and  a  predilection  for  fighting  created  a 
bad  impression;  his  father  was  overheard  to  say  (pray, 
according  to  one  account)  that  "  if  it  pleased  God  to  take 
one  of  his  children,  he  could  best  spare  Isaac."  Later,  he 
seems  to  have  turned  over  a  new  leaf,  and  in  1643  we  find 
him  entered  at  St  Peter's  College,  Cambridge,  and  afterwards 
at  Trinity.  Having  now  become  exceedingly  studious,  he 
made  considerable  progress  in  literature,  natural  philosophy, 
anatomy,  botany,  and  chemistry  —  the  three  last  with  a 
view  to  medicine  as  a  profession, — and  later  in  chronology, 
geometry,  and  astronomy.  He  then  proceeded  on  a  sort 
of  "Grand  Tour"  through  France,  Italy,  to  Smyrna,  Con- 
stantinople, back  to  Venice,  and  then  home  through  Germany 
and  Holland.  His  stay  in  Constantinople  had  a  great 
influence  on  his  after  life ;  for  he  here  studied  the  works  of 
Chrysostom,  and  thus  had  his  thoughts  turned  to  divinity. 
But  for  this,  his  great  advance  on  the  work  of  his  pre- 
decessors in  the  matter  of  the  infinitesimal  calculus  might 
have  been  developed  to  such  an  extent  that  the  name  of 
Barrow  would  have  been  inscribed  on  the  roll  of  fame  as 
at  least  the  equal  of  his  mighty  pupil  Newton. 

Immediately  on  his  return  to  England  he  was  ordained, 
and  a  year  later,  at  the  age  of  thirty,  he  was  appointed  to 
the  Greek  professorship  at  Cambridge ;  his  inaugural  lectures 
were  on  the  subject  of  the  Rhetoric  of  Aristotle,  and  this 
choice  had  also  a  distinct  effect  on  his  later  mathematical 
work.  In  1662,  two  years  later,  he  was  appointed  Professor 
of  Geometry  in  Gresham  College ;  and  in  the  following  year 
he  was  elected  to  the  Lucasian  Chair  of  Mathematics,  just 
founded  at  Cambridge.  This  professorship  he  held  for  five 
years,  and  his  office  created  the  occasion  for  his  Lectiones 
Mathematics,  which  were  delivered  in  the  years  16^4-5-6 
(Habita  Cantabrigice).  These  lectures  were  published, 
according  to  Prof.  Benjamin  Williamson  (Encyc.  Brit. 


INTRODUCTION  7 

(Times  edition).  Art.  on  Infinitesimal  Calculus)  in  1670; 
this,  however,  is  wrong :  they  were  not  published  until 
1683,  under  the  title  of  Lectiones  Mathematics.  What  was 
published  in  1670  was  the  Lectiones  Opticce  et  Geometries ; 
the  Lectiones  Mathematics  were  philosophical  lectures  on 
the  fundamentals  of  mathematics  and  did  not  have  much 
bearing  on  the  infinitesimal  calculus.  They  were  followed 
by  the  Lectiones  Optics  and  lectures  on  the  works  of 
Archimedes,  Apollonius,  and  Theodosius;  in  what  order 
these  were  delivered  in  the  schools  of  the  University  I  have 
been  unable  to  find  out ;  but  the  former  were  published  in 
1669,  "Imprimatur"  having  been  granted  in  March  1668, 
so  that  it  was  probable  that  they  were  the  professorial 
lectures  for  1667  ;  thus  the  latter  would  have  been  delivered 
in  1668,  though  they  were  not  published  until  1675,  and  then 
probably  by  Collins.  The  great  work,  Lectiones  Geometricce^ 
did  not  appear  as-  a  separate  publication  at  first :  as  stated 
above,  it  was  issued  bound  up  with  the  second  edition  of 
the  Lectiones  Opticce;  and,  judging  from  the  fact  that  there 
does  not,  according  to  the  above  dates,  appear  to  have  been 
any  time  for  their  public  delivery  as  Lucasian  Lectures, 
since  Imprimatur  was  granted  for  the  combined  edition  in 
1669;  also  from  the  fact  that  Barrow's  Preface  speaks  of 
six  out  of  the  thirteen  lectures  as  "  matters  left  over  from 
the  Optics,"  which  he  was  induced  to  complete  to  form  a 
separate  work;  also  from  the  most  .conclusive  fact  of  all, 
that  on  the  title-page  of  the  Lectiones  Geometricce  there  is  no 
mention  at  all  of  the  usual  notice  "  Habitae  Cantabrigise  " ; — 
judging  from  these  facts,  I  do  not  believe  that  the  "Lectiones 
Geometric^"  were  delivered  as  Lucasian  Lectures.  Should  this 
be  so,  it  would  clear  up  a  good  many  difficulties ;  it  would 
corroborate  my  suggestion  that  they  were  for  a  great  part 
evolved  during  his  professorship  at  Gresham  College ;  also 
it  would  make  it  almost  certain  that  they  would  have  been 
given  as  internal  college  lectures,  and  that  Newton  would 
have  heard  them  in  1663-1664. 

Now,  it  was  in  1664  that  Barrow  first  came  into  close 
personal  contact  with  Newton ;  for  in  that  year,  he 
examined  Newton  in  Euclid,  as  one  of  the  subjects  for 
a  mathematical  scholarship  at  Trinity  College,  of  which 
Newton  had  been  a  subsizar  for  three  years  ;  and  it  was  due 


8      BARROWS  GEOMETRICAL  LECTURES 

to  Barrow's  report  that  Newton  was  led  to  study  the 
Elements  more  carefully  and  to  form  a  better  estimate  of 
their  value.  The  connection  once  started  must  have 
developed  at  a  great  pace,  for  not  only  does  Barrow  secure 
the  succession  of  Newton  to  the  Lucasian  chair,  when  he 
relinquished  it  in  1669,  but  he  commits  the  publication 
of  his  Lectiones  Optictz  to  the  foster  care  of  Newton  and 
Collins.  He  himself  had  now  determined  to  devote  the 
rest  of  his  life  to  divinity  entirely;  in  1670  he  was  created 
a  Doctor  of  Divinity,  in  1672  he  succeeded  Dr  Pearson  as 
Master  of  Trinity,  in  1675  ne  was  chosen  Vice-Chancellor 
of  the  University;  and  in  1677  he  died,  and  was  buried 
in  Westminster  Abbey,  where  a  monument,  surmounted  by 
his  bust,  was  soon  afterwards  erected  to  his  memory  by 
his  friends  and  admirers. 

Ill 
THE   WORKS    OF    BARROW 

Barrow  was  a  very  voluminous  writer.  On  inquiring  of 
the  Librarian  of  the  Cambridge  University  Library  whether 
he  could  supply  me  with  a  complete  list  of  the  works  of 
Barrow  in  order  of  publication,  I  was  informed  that  the 
complete  list  occupied  four  columns  in  the  British  Museum 
Catalogue  \  This  of  course  would  include  his  theological 
works,  the  several  different  editions,  and  the  translations  of 
his  Latin  works.  The  following  list  of  his  mathematical 
works,  such  as  are  important  for  the  matter  in  hand  only, 
is  taken  from  the  Bibliotheca  Britannica  (by  Robert  Watt, 
Edinburgh,  1824) : — 

1.  Euclid's  Elements,  Camb.  1655. 

2.  Euclid's  Data,  Camb.  1657. 

3.  Lectiones  Opffcorum  Phenomenon,  Lond.  1669. 

4.  Lectiones  Opticczet  Geometric^,       Lond.  1670 

(in  2  vols.,  1674;  trans.,  Edmond  Stone,  1735). 

5.  Lectiones  Mathematics,  Lond.  1683. 
This  list  makes  it  absolutely  certain  that  Williamson  is 

wrong  in  stating  that  the  lectures  in  geometry  were  published 
under  the  title  of  "  Mathematical  Lectures."  This,  how- 
ever, is  not  of  much  consequence ;  the  important  point  in 


INTRODUCTION  9 

the  list,  assuming  it  to  be  perfectly  correct  as  it  stands,  is 
that  the  lectures  on  Optics  were  first  published  separately 
in  1669.  1°  tne  following  year  they  were  reissued  in  a 
revised  form  with  the  addition  of  the  lectures  on  geometry. 

The  above  books  were  all  in  Latin  and  have  been 
translated  by  different  people  at  one  time  or  another. 

IV 
ESTIMATE   OF   BARROW'S   GENIUS 

The  writer  of  the  article  on  "  Barrow,  Isaac,"  in  the  ninth 
(Times]  edition  of  the  Encyclopedia  Britannica,  from  which 
most  of  the  details  in  Section  II  have  been  taken,  remarks  : — 

''By  his  English  contemporaries  Barrow  was  considered 
a  mathematician  second  only  to  Newton.  Continental 
writers  do  not  place  him  so  high,  and  their  judgment  is  prob- 
ably the  more  correct  one" 

Founding  my  opinion  on  the  Lectiones  Geometricce  alone,  I 
fail  to  see  the  reasonableness  of  the  remark  I  have  italicized. 
Of  course,  it  was  only  natural  that  contemporary  continental 
mathematicians  should  belittle  Barrow,  since  they  claimed 
for  Fermat  and  Leibniz  the  invention  of  the  infinitesimal 
calculus  before  Newton,  and  did  not  wish  to  have  to  con- 
sider in  Barrow  an  even  prior  claimant.  We  see  that  his 
own  countrymen  placed  him  on  a  very  high  level;  'and 
surely  the  only  way  to  obtain  a  really  adequate  opinion  of  a 
scientist's  worth  is  to  accept  the  unbiased  opinion  that  has 
been  expressed  by  his  contemporaries,  who  were  aware  of 
all  the  facts  and  conditions  of  the  case ;  or,  failing  that,  to 
try  to  form  an  unbiased  opinion  for  ourselves,  in  the  position 
of  his  contemporaries.  An  obvious  deduction  may  be  drawn 
from  the  controversy  between  Newton  and  Hooke ;  the 
opinion  of  Barrow's  own  countrymen  would  not  be  likely  to 
err  on  the  side  of  over-appreciation,  unless  his  genius  was 
great  enough  to  outweigh  the  more  or  less  natural  jealousy 
that  ever  did  and  ever  will  exist  amongst  great  men  occupied 
on  the  same  investigations.  Most  modern  criticism  of 
ancient  writers  is  apt  to  fail,  because  it  is  in  the  hands  of 
the  experts;  perhaps  to  some  degree  this  must  be  so,  yet 
you  would  hardly  allow  a  K.C.  to  be  a  fitting  man  for  a  jury. 


io  BARROW'S  GEOMETRICAL  LECTURES 

Criticism  by  experts,  unless  they  are  themselves  giants  like 
unto  the  men  whose  works  they  criticize,  compares,  perhaps 
unconsciously,  their  discoveries  with  facts  that  are  now 
common  knowledge,  instead  of  considering  only  and  solely 
the  advance  made  upon  what  was  then  common  knowledge. 
Thus  the  skilled  designers  of  the  wonderful  electric  engines 
of  to-day  are  but  as  pigmies  compared  with  such  giants  as 
a  Faraday. 

Further,  in  the  case  of  Barrow,  there  are  several  other 
things  to  be  taken  into  account.  We  must  consider  his  dis- 
position, his  training,  his  changes  of  intention  with  regard 
to  a  career,  the  accident  of  his  connection  with  such  a  man 
as  Newton,  the  circumstances  brought  about  by  the  work  of 
his  immediate  predecessors,  and  the  ripeness  of  the  time 
for  his  discoveries. 

His  disposition  was  pugnacious,  though  not  without  a 
touch  of  humour ;  there  are  many  indications  in  the  Lectiones 
Geometries  alone  of  an  inclination  to  what  I  may  call,  for 
lack  of  a  better  term,  a  certain  contributory  laziness ;  in  this 
way  he  was  somewhat  like  Fermat,  with  his  usual  "  I  have 
got  a  very  beautiful  proof  of  .  .  .  :  if  you  wish,  I  will  send 
it  to  you ;  but  I  dare  say  you  will  be  able  to  find  it  for  your- 
self"; many  of  Barrow's  most  ingenious  theorems,  one  or 
two  of  his  most  far-reaching  ones,  are  left  without  proof, 
though  he  states  that  they  are  easily  deduced  from  what  has 
:  gone  before.  He  evidently  knows  the  importance  of  his 
discoveries ;  in  one  place  he  remarks  that  a  certain  set  of 
theorems  are  a  "mine  of  information,  in  which  should  any- 
one investigate  and  explore,  he  will  find  very  many  things 
r.of  this  kind.  Let  him  do  so  who  must,  or  if  it  pleases  him." 
He  omits  the  proof  of  a  certain  theorem,  which  he  states 
has  been  very  useful  to  him  repeatedly ;  and  no  wonder  it 
has,  for  it  turns  out  to  be  the  equivalent  to  the  differentia- 
tion of  a  quotient  •  and  yet  he  says,  "It  is  sufficient  for  me 
to  mention  this,  and  indeed  I  intend  to  stop  here  for  a  while." 
It  is  not  at  all  strange  that  the  work  of  such  a  man  should 
come  *to  be  underrated. 

His  pugnacity  is  shown  in  the  main  object  pervading  the 
whole  of  the  Lectiones  Geometries  •  he  sets  out  with  the  one 
express  intention  of  simplifying  and  generalizing  the  existing 
methods  of  drawing  tangents  to  curves  of  all  kinds  and  of 


INTR  OD  UCTION  \  i 

finding  areas  and  volumes ;  there  is  distinct  humour  in  his 
glee  at  "wiping  the  eye"  of  some  other  geometer,  ancient 
or  modern,  whose  solution  of  some  particular  problem  he 
has  not  only  generalized  but  simplified. 

"Gregory  St  Vincent  gave  this,  but  (if  I  remember 
rightly)  proved  with  wearisome  prolixity." 

"Hence  it  follows  immediately  that  all  curves  of  this 
kind  are  touched  at  any  one  point  by  one  straight  line 
only.  .  .  .  Euclid  proved  this  as  a  special  case  for  the 
circle,  Apollonius  for  the  conic  sections,  and  other  persons 
in  the  case  of  other  curves." 

His  early  training  was  promiscuous,  and  could  have  had 
no  other  effect  than  to  have  fostered  an  inclination  to  leave 
others  to  finish  what  he  had  begun.  His  Greek  professor-"1 
ship  and  his  study  of  Aristotle  would  tend  to  make  him-a 
confirmed  geometrician,  revelling  in  the  "elegant  solution" 
and  more  or  less  despising  Cartesian  analysis  because  of 
its  then  (frequently)  cumbersome  work,  and  using  it  only 
with  certain  qualms  of  doubt  as  to  its  absolute  rigour. 
For  instance,  he  almost  apologizes  for  inserting,  at  the 
very  end  of  Lecture  X,  which  ends  the  part  of  the  work 
devoted  to  the  equivalent  of  the  differential  calculus,  his 
"  a  and  e  "  method — the  prototype  of  the  "  h  and  k  "  method 
of  the  ordinary  text-books  of  to-day. 

Another  light  is  thrown  on  the  matter  of  Cartesian 
geometry,  or  rather  its  applications,  by  Lecture  VI ;  in  this, 
for  the  purpose  of  establishing  lemmas  to  be  used  later, 
Barrow  gives  fairly  lengthy  proofs  that 

(i)  my±xy  =  mx2/fr,     (2)    ±yx+gx  —  my  =  mx2/r 

represent  hyperbolas,  instead  of  merely  stating  the  fact  on 
account  of  the  factorizing  of  mx^/b  +  xy,  mx^jr±xy.  The1 
lengthiness  of  these  proofs  is  to  a  great  extent  due  to  the 
fact  that,  although  the  appearance  of  the  work  is  algebraical, 
the  reasoning  is  almost  purely  geometrical.  It  is  also  to  be 
noted  that  the  index  notation  is  rarely  used,  at  least  not  till 
very  late  in  the  book  in  places  where  he  could  do  nothing  else, 
although  Wallis  had  used  even  fractional  indices  a  dozen 
years  before.  In  a  later  lecture  we  have  the  truly  terrifying 
equation  (rrkk  -  rrff+  2/mpa)lkk  =  (rrmm  +  ^fmpd)lkk. 
Again  we  must  note  the  fact  that  all  Barrow's  work, 


12   BARROW'S   GEOMETRICAL  LECTURES 

without  exception,  was  geometry,  although  it  is  fairly  evident 
that  he  used  algebra  for  his  own  purposes. 

From  the  above,  it  is  quite  easy  to  see  a  reason  why 
Barrow  should  not  have  turned  his  work  to  greater  account ; 
but  in  estimating  his  genius  one  must  make  allowance  for 
this  disability  in,  or  dislike  for,  algebraic  geometry,  read 
into  his  work  what  could  have  been  got  out  of  it  (what  I 
am  certain  both  Newton  and  Leibniz  got  out  of  it),  and 
not  ,stop  short  at  just  what  was  actually  published.  It 
must  chiefly  be  remembered  that  these  old  geometers  could 
use  their  geometrical  facts  far  more  readily  and  surely  than 
many  mathematicians  of  the  present  day  can  use  their 
analysis.  As  a  justification  of  the  extremely  high  estimate 
I  have  formed,  from  the  Lectiones  Geometries  alone,  of 
Barrow's  genius,  I  call  the  attention  of  the  reader  to  the 
list  of  analytical  equivalents  of  Barrow's  theorems  given  on 
page  30,  if  he  has  not  the  patience  to  wade  through  the 
running  commentary  which  stands  instead  of  a  full  trans- 
lation of  this  book. 

V 
THE   SOURCES   OF   BARROW'S   IDEAS 

There  is  too  strong  a  resemblance  between  the  methods 
to  leave  room  for  doubt  that  Barrow  owed  much  of  his  idea 
of  integration  to  Galileo  and  Cavalieri  (or  Roberval,  if  you 
will).  On  the  question  as  to  the  sources  from  which  he 
derived  his  notions  on  differentiation  there  is  considerably 
more  difficulty  in  deciding;  and  the  comparatively  narrow 
range  of  my  reading  makes  me  diffident  in  writing  anything 
that  may  be  considered  dogmatic  on  this  point ;  and  yet  if 
I  do  not  do  so,  I  shall  be  in  danger  of  not  getting  a  fair 
hearing.  The  following  remarks  must  therefore  be  con- 
sidered in  the  nature  of  the  plea  of  a  "counsel  for  the 
defence,"  who  believes  absolutely  in  his  client's  case ;  or  as 
suggestions  that  possibly,  even  if  not  probably,  come  very 
near  to  the  truth. 

The  general  opinion  would  seem  to  be  that  Barrow  was  a 
mere  improver  on  Fermat.  But  Barrow  was  conscientious  to 
a  fault ;  and  if  we  are  to  believe  in  his  honesty,  the  source 
of  his  ideas  could  not  have  been  Fermat.  For  Barrow  re- 


INTRODUCTION  13 

ligiously  gives  references  to  the  ancient  and  contemporary 
mathematicians  whose  work  he  quotes.  These  references 
include  Cartesius,  Hugenius,  Galilseus,  Gregorius  a  St  Vin- 
centio,  Gregorius  Aberdonensis,  Wallis,  and  many  others, 
with  Euclides,  Aristoteles,  Archimedes,  Apollonius,  among 
the  ancients;  but,  as  far  as  I  can  find,  no  mention  is  made1 
of  Fermat  in  any  place ;  nor  does  Barrow  use  Fermat's  idea 
of  determining  the  tangent  algebraically  by  consideration  of 
a  maximum  or  minimum  ;  these  points  entirely  contradict 
the  notion  that  he  was  a  mere  improver  on  Fermat,  which 
seems  to  have  arisen  because  Barrow  uses  the  same  letter, 
e,  for  his  increment  of  x,  and  only  adds  another,  a,  to  signify 
the  increment  of  y.  I  suggest  that  this  was  only  a  coinci- 
dence ;  that  both  adopted  the  letter  e  (Fermat  seems  to  have 
used  the  capital  E)  as  being  the  initial  letter  of  the  word  excess, 
whilst  Barrow  in  addition  used  the  letter  a,  the  initial  letter 
of  the  word  additional;  if  he  was  a  mere  improver  on  Fermat, 
the  improvement  was  a  huge  one,  for  it  enabled  Barrow  to 
handle,  without  the  algebraical  difficulties  of  Fermat,  im- 
plicit functions  as  well  as  explicit  functions.  On  the  other 
hand,  Barrow  may  have  got  the  notion  of  using  arithmetic 
and  geometric  means,  with  which  he  performs  some  wonders, 
from  Fermat,  who  apparently  was  the  first  to  use  them,  though 
by  Barrow's  time  they  were  fairly  common  property,  being 
the  basis  of  all  systems  of  logarithms  ;  and  Barrow's  con- 
dition of  tangency  was  so  similar  to  the  method  of  Fermat 
that,  while  he  could  not  very  well  use  any  other  condition 
with  his  choice  of  the  definition  of  a  tangent,  Barrow  may 
have  deliberately  omitted  any  reference  to  Fermat,  for  fear 
that  thereby  he  might,  by  the  reference  alone,  provoke  ac- 
cusations of  plagiarism.  As  I  have  already  remarked,  there  ""* 
is  a  distinct  admiration  for  the  work  of  Galileo,  and  the 
idea  of  time  as  the  independent  variable  obsesses  the  first 
few  lectures  ;  however,  he  simply  intended  this  as  a  criterion 
by  means  of  which  he  could  be  sure  that  one  of  his  variables 
increased  uniformly,/  or  in  certain  of  his  theorems  in  the 
later  parts  that  he  might  consider  hisjF  as  a  function  of  a 
function  ;  but  in  most  of  the  later  lectures  the  idea  of  time 
becomes  quite  insignificant.  This  is,  of  course,  explained  by 
the  fact  that  the  original  draft  of  the  geometrical  lectures 
consisted  only  of  the  lectures  numbered  VI  to  XII  (includ- 


14  BARROW'S   GEOMETRICAL   LECTURES 

ing  the  appendices,  with  the  possible  exception  of  Appendix 
3  to  Lect.  XII) ;  for  we  read  in  the  Preface  that  Barrow 
"  falling  in  with  his  wishes  (I  will  not  say  very  willingly)  added 
the  first  five  lectures."  The  word  "  his  "  refers  to  Librarius, 
which — for  lack  of  a  better  word,  or  of  editor,  which  I  do 
not  like — I  have  translated  as  the  publisher-,  but  I  think  it 
refers  to  Collins,  for,  in  Barrow's  words,  "  John  Collins  looked 
after  the  publication." 

The  opinion  I  have  formed  is  that  the  idea  of  the  differ- 
ential triangle,  upon  which  all  attention  seems  (quite 
wrongly)  to  be  focussed,  when  considering  the  work  of 
Barrow,  was  altogether  his  own  original  concept  •  and  to  call 
it  a  mere  improvement  on  Fermat's  method,  in  that  he  uses 
two  increments  instead  of  one,  is  absurd.  The  discovery 
was  the  outcome  of  Barrow's  definition  of  a  tangent,  wholly 
and  solely ;  and  the  method  of  Fermat  did  not  consider  this. 

The  mental  picture  that  I  form  of  Barrow,  and  of  the 
events  that  led  to  this  discovery,  amongst  others  far  more 
important,  is  that  of  the  Professor  of  Geometry  at  Gresham 
College,  who  has  to  deliver  lectures  on  his  subject; 
he  reads  up  all  that  he  can  lay  his  hands  on,  decides 
that  it  is  all  very  decent  stuff  of  a  sort,  yet  pugnaciously 
determines  that  he  can  and  will  "go  one  better."  In  the 


Fig.  A. 


Fig.  B. 


course  of  his  researches,  he  is  led  from  one  thing  to 
another  until  he  comes  to  the  paraboliform  construction  of 
Lect.  IX,  §  4,  perceives  its  usefulness  and  inner  meaning, 
and  immediately  conceives  the  idea  of  the  differential 
triangle.  I  think  if  any  reader  compares  the  two  figures 
above,  Fig.  A  used  for  his  construction  of  the  paraboliforms, 
Fig.  B  for  the  differential  triangle,  he  will  no  longer  inquire 


INTRODUCTION  15 

for  the  source  of  Barrow's  idea,  unless  perhaps  he   may 
prefer  to  refer  it  to  Lect.  X,  §  1 1. 

Personally,  I  have  no  doubt  that  it  was  a  flash  of  inspira- 
tion, suggested  by  the  first  figure  ;  and  that  it  was  Barrow's 
luck  to  have  first  of  all  had  occasion  to  draw  that  figure,  and 
secondly  to  have  had  the  genius  to  note  its  significance  and 
be  able  to  follow  up  the  clue  thus  afforded.  As  further,, 
corroborative  evidence  that  Barrow's  ideas  were  in  the  main 
his  own  creations,  we  have  the  facts  that  he  was  alone  in 
using  habitually  the  idea  of  a  curve  being  a  succession  of 
an  infinite  number  of  infinitely  short  straight  lines,  the  pro- 
longation of  any  one  representing  the  tangent  at  the  point 
on  the  curve  for  which  the  straight  line,  or  either  end  of 
of  it,  stood ;  also  that  he  could  not  see  any  difference 
between  indefinitely  narrow  rectangles  and  straight  lines  as 
the  constituents  of  an  area.  If  his  methods  had  required  it, 
which  they  did  not,  he  would  no  doubt  have  proved  rigor- 
ously that  the  error  could  be  made  as  small  as  he  pleased 
by  making  the  number  of  parts,  into  which  he  had  divided 
his  area,  large  enough;  this  was  indeed  the  substance 
of  Pascal's  defence  of  Cavalieri's  method  of  "indivisibles," 
and  the  idea  is  used  in  Lect.  XII,  App.  II,  §  6. 

I  have  remarked  that,  in  considering  the  work  of  Barrow, 
all  attention  seems  to  be  quite  wrongly  focussed  on  the  differ- 
ential triangle.  I  hope  to  convince  readers  of  this  volume 
that  the  differential  triangle  was  only  an  important  side- 
issue  in  the  Lectiones  Geometrical;  certainly  Barrow  only 
considered  it  as  such.  Barrow  really  had,  concealed  under 
the  geometrical  form  that  was  his  method,  a  complete  treatise 
on  the  elements  of  the  calculus. 

The  question  may  then  be  asked  why,  if  all  this  is  true, 
did  Barrow  not  finish  the  work  he  had  begun;  and  the 
answer,  I  take  it,  is  inseparably  bound  up  with  the  peculiar 
disposition  of  Barrow,  his  growing  desire  to  forsake 
mathematics  for  divinity,  and  the  accident  of  having  first 
as  his  pupil  and  afterwards  as  his  co-worker,  and  one  in 
close  personal  contact  with  him,  a  man  like  Newton,  whose 
analytical  mind  was  so  peculiarly  adapted  to  the  task  of 
carrying  to  a  successful  conclusion  those  matters  which 
Barrow  saw  could  not  be  developed  to  anything  like  the 
extent  by  his  own  geometrical  method.  One  writer  has 


1 6    BARROW'S   GEOMETRICAL  LECTURES 

stated  that  the  great  genius  of  Barrow  must  be  admitted, 
if  only  for  the  fact  that  he  recognized  in  the  early  days  of 
Newton's  career  the  genius  of  the  man,  his  pupil,  that  was 
afterwards  to  overshadow  him.  Also,  if  I  fail  to  make 
out  my  contention  that  Barrow's  ideas  were  in  the  main 
original,  the  same  remark  can  with  justice  be  applied  to 
him  that  William  Wallace  in  similar  circumstances  applied 
to  Descartes,  that  if  it  were  true  that  he  borrowed  his  ideas 
on  algebra  from  others,  this  fact,  "  would  only  illustrate 
the  genius  of  the  man  who  could  pick  out  from  other  works 
all  that  was  productive,  and  state  it  with  a  lucidity  that  makes 
it  look  his  own  discovery  " ;  for  the  lucidity  is  there  all  right 
in  this  work  of  Barrow,  only  it  wants  translating  into  analy- 
tical language  before  it  can  be  readily  grasped  by  anyone 
but  a  geometer. 

VI 

MUTUAL   INFLUENCE   OF   NEWTON   AND 
BARROW 

I  can  image  that  Barrow's  interest,  as  a  confirmed 
geometer,  would  have  been  first  aroused  by  >oung  Newton's 
poor  show  in  his  scholarship  paper  on  Euclid.  This  was  in 
April  1664,  the  year  of  the  delivery  of  Barrow's  first  lectures 
as  Lucasian  Professor,  and,  according  to  Newton's  own 
words,  just  about  the  time  that  he,  Newton,  discovered  his 
method  of  infinite  series,  led  thereto  by  his  reading  of  the 
works  of  Descartes  and  Wallis.  Newton  no  doubt  attended 
these  lectures  of  Barrow,  and  the  probability  is  that  he 
would  have  shown  Barrow  his  work  on  infinite  series ;  for 
this  would  seem  to  have  been  the  etiquette  or  custom  of 
the  time;  for  we  know  that  in  1669  Newton  communicated 
to  Collins  through  Barrow  a  compendium  of  his  work  on 
fluxions  (note  that  this  is  the  year  of  the  preparation  for 
press  of  the  Lectiones  Optica  et  Geometric^.  Barrow  could 
not  help  being  struck  by  the  incongruity  (to  him)  of  a  man 
of  Newton's  calibre  not  appreciating  Euclid  to  the  full ;  at 
the  same  time  the  one  great  mind  would  be  drawn  to  the 
other,  and  the  connection  thus  started  would  have  ripened 
inevitably.  I  suggest  as  a  consequence  that  Barrow  would 
show  Newton  his  own  geometry,  Newton  would  naturally 


INTRODUCTION  17 

ask  Barrow  to  explain  how  he  had  got  the  idea  for  some  of 
his  more  difficult  constructions,  and  Barrow  would  let  him 
into  the  secret.  "  I  find  out  the  constructions  by  this  little 
list  of  rules,  and  methods  for  combining  them."  "  But,  my 
dear  sir,  the  rules  are  far  more  valuable  than  the  mere  find- 
ing of  the  tangents  or  the  areas."  "  All  right,  my  boy,  if  you 
think  so,  you  are  welcome  to  them,  to  make  what  you  like  of, 
or  what  you  can ;  only  do  not  say  you  got  them  from  me,  I'll 
stick  to  my  geometry."*  This  was  probably  the  occasion  when 
Newton  persuaded  Barrow  that  the  differential  triangle  was 
more  general  than  all  his  other  theorems  put  together ;  also 
later  when  the  Geometry  was  being  got  ready  for  press, 
Newton  probably  asked  Barrow  to  produce  from  his  stock 
of  theorems  others  necessary  to  complete  his,  Barrow's, 
Calculus,  the  result  being  the  appendices  to  Lect.  XII. 

The  rest  of  the  argument  is  a  matter  of  dates.  Barrow 
was  Professor  of  Greek  from  1660  to  1662,  then  Professor 
of  Geometry  at  Gresham  College  from  1662  to  1664,  and 
Lucasian  Professor  from  1664  to  1669;  Newton  was  a 
member  of  Trinity  College  from  1661,  and  was  in  residence 
until  he  was  forced  from  Cambridge  by  the  plague  in  the 
summer  of  1665  ;  from  manuscript  notes  in  Newton's  hand- 
writing, it  was  probably  during,  and  owing  to,  this  enforced 
absence  from  Cambridge  (and,  I  suggest,  away  from  the 
geometrical  influence  of  Barrow)  that  he  began  to  develop 
the  method  of  fluxions  (probably  in  accordance  with  some 
such  permission  from  Barrow  as  that  suggested  in  the  purely 
imaginative  interview  above). 

The  similarity  of  the  two  methods  of  Barrow  and  Newton 
is  far  too  close  to  admit  of  them  being  anything  else  but 
the  outcome  of  one  single  idea;  and  I  argue  from  the  dates 
given  above  that  Barrow  had  developed  most  of  his  geometry 
from  the  researches  begun  for  the  necessities  of  lectures  at 
Gresham  College.  We  know  that  Barrow's  work  on  the 
difficult  theorems  and  problems  of  Archimedes  was  largely 
a  suggestion  of  a  kind  of  analysis  by  which  they  were  reduced 
to  their  simple  component  problems.  What  is  then  more 
likely  than  that  this  is  an  intentional  or  unintentional  crypto- 
grammatic  key  to  Barrow's  own  method?  I  suggest  that  it 

*  Of  course  this  is  imaginative  retrospective  prophecy  ;  I  beg  that  no 
one  will  take  the  inverted  commas  to  signify  quotations. 

2 


1 8    BARROWS   GEOMETRICAL  LECTURES 

is  more  than  likely, — IT  is.     As  I  said,  the  similarity  of  the 
two  methods  of  Newton  and  Barrow  is  very  striking. 
For  the  fluxional  method  the  procedure  is  as  follows : — 

(1)  Substitute  x  +  xo  for  x  and  y+yo  for  y  in  the  given 
equation  containing  the  fluents  x  and  y. 

(2)  Subtract  the  original  equation,  and  divide  through  by  o\ 

(3)  Regard  o  as  an  evanescent  quantity,  and  neglect  o 
and  its  powers. 

Barrow's  rules,  in  altered  order  to  correspond,  are : — 

(2)  After  the  equation  has  been  formed  (Newton's  rule  i), 
reject  all  terms  consisting  of  letters  denoting  constant  or 
determined  quantities,  or  terms  which  do  not  contain  a  or  e 
(which  are  equivalent  to  Newton's yo  and  xo  respectively) ; 
for  these  terms  brought  over  to  one  side  of  the  equation  will 
always  be  equal  to  zero  (Newton's  rule  2,  first  part). 

(i)  In  the  calculation,  omit  all  terms  containing  a  power 
of  a  or  e,  or  products  of  these  letters ;  for  these  are  of  no 
value  (Newton's  rule  2,  second  part,  and  rule  3). 

(3)  Now  substitute  m  (the  ordinate)  for  a,  and  t  (the  sub- 
tangent)  for  e.     (This  corresponds  with  Newton's  next  step, 
the  obtaining  of  the  ratio  x  :  y,  which  is  exactly  the  same  as 
Barrow's  e  :  a. ) 

The  only  difference  is  that  Barrow's  way  is  the  more  suited 
to  his  geometrical  purpose  of  finding  the  "  quantity  of  the 
subtangent,"  and  Newton's  method  is  peculiarly  adapted  to 
analytical  work,  especially  in  problems  on  motion.  Barrow 
left  his  method  as  it  stood,  though  probably  using  it  freely 
(mark  the  word  usitatum  on  page  119,  which  is  a  frequentative 
derivative  of  utor,  I  use)  to  obtain  hints  for  his  tangent 
problems,  but  not  thinking  much  of  it  as  a  method  compared 
with  a  strictly  geometrical  method ;  yet  admitting  it  into 
his  work,  on  the  advice  of  a  friend,  on  account  of  its 
generality.  On  the  other  hand,  Newton  perceived  at  once 
the  immense  possibilities  of  the  analytical  methods  intro- 
duced by  Descartes,  and  developed  the  idea  on  his  own 
lines,  to  suit  his  own  purposes. 

There  is  still  another  possibility.  In  the  Preface  to  the 
Optics,  we  read  that  "  as  delicate  mothers  are  wont,  I  com- 
mitted to  the  foster  care  of  friends,  not  unwillingly,  my  dis- 
carded child"  .  .  .  These  two  friends  Barrow  mentions  by 
name:  "Isaac  Newton  ...  (a  man  of  exceptional  ability 


INTRODUCTION  19 

and  remarkable  skill)  has  revised  the  copy,  warning  me  of 
many  things  to  be  corrected,  and  adding  some  things  from 
his  own  work."  .  .  .  Newton's  additions  were  probably  con- 
fined to  a  great  extent  to  the  Optics  only  ;  but  the  geometrical 
lectures  (seven  of  them  at  least)  were  originally  designed  as 
supplementary  to  the  Optics,  and  would  be  also  looked  over 
by  Newton  when  the  combined  publication  was  being  pre- 
pared. .  .  .  "John  Collins  has  attended  to  the  publication." 
Hence,  it  is  just  possible  that  Newton  showed  Barrow  his 
method  of  fluxions  first,  and  Barrow  inserted  it  in  his  own 
way  ;  this  supposition  would  provide  an  easy  explanation 
of  the  treatment  accorded  to  the  batch  of  theorems  that 
form  the  third  appendix  to  Lect.  XII  ;  they  seem  to  be 
hastily  scrambled  together,  compared  with  the  orderly  treat- 
ment of  the  rest  of  the  book,  and  are  without  demonstration  ; 
and  this,  although  they  form  a  necessary  complement  for 
the  completion  of  the  standard  forms  and  rules  of  procedure. 
I  say  that  this  is  possible,  but  I  do  not  think  it  is  at  all 
probable ;  for  it  is  to  be  noted  that  Barrow's  description 
of  the  method  is  in  the  first  person  singtdar  (although,  when 
giving  the  reason  for  its  introduction,  he  says  "frequently 
used  by  us");  and  remembering  the  authentic  accounts  of 
Barrow's  conscientious  honesty,  and  also  judging  by  the 
later  work  of  Newton,  I  think  that  the  only  alternative  to 
be  considered  is  that  first  given.  Also,  if  that  is  accepted, 
we  have  a  natural  explanation  of  the  lack  of  what  I  call  the 
true  appreciation  of  Barrow's  genius.  Barrow  could  see  the 
limitations  imposed  by  his  own  geometrical  methods  (none 
so  well  as  he,  naturally,  being  probably  helped  to  this  con- 
clusion by  his  discussions  with  Newton);  he  felt  that  the 
correct  development  of  his  idea  was  on  purely  analytical 
lines,  he  recognised  his  own  disability  in  that  direction  and 
the  peculiar  aptness  of  Newton's  genius  for  the  task,  and, 
lastly,  the  growing  desire  to  forsake  mathematics  for  divinity 
made  him  only  too  willing  to  hand  over  to  the  foster  care 
of  Newton  and  Collins  his  discarded  child  "  to  be  led  out 
and  set  forth  as  might  seem  good  to  them!''  "  Carte  blanche  " 
of  such  a  sweeping  character  very  often  has  exactly  the 
opposite  effect  to  that  which  is  intended ;  and  so  probably 
Newton  and  Collins  forbore  to  make  any  serious  alterations 
or  additions,  out  of  respect  for  Barrow;  for  although  the 


20    BARROW'S  GEOMETRICAL  LECTURES 

allusion  to  the  revision  properly  applies  only  to  the  Optics, 
it  may  fairly  be  assumed  that  it  would  be  extended  to  the 
Geometry  as  well ;  and  if  not  then,  at  any  rate  later,  for, 
quoting  a  quotation  by  Canon  Overton  in  the  Dictionary 
of  National  Biography  (source  of  the  quotation  not  stated), 
which  refers  to  Barrow's  pique  at  the  poor  reception  that 
was  accorded  to  the  geometrical  lectures — and  does  not  this 
show  the  high  opinion  that  Barrow  had  of  them  himself,  and 
lend  colour  to  my  suggestion  that  they  were  never  delivered  as 
Lucasian  Lectures?;  also  note  his  remark  in  the  Preface,  given 
later,  "  The  other  seven,  as  I  said,  I  expose  more  freely  to  your 
view,  hoping  that  there  is  nothing  in  them  that  it  will  displease 
the  erudite  to  see" — "  When  they  had  been  some  time  in  the 
world,  having  heard  of  a  very  few  who  had  read  and  considered 
them  thoroughly,  the  little  relish  that  such  things  met  with 
helped  to  loose  him  more  from  those  speculations  and  heighten 
his  attention  to  the  studies  of  morality  and  divinity."  Does 
not  this  read  like  the  disgust  at  people  forsaking  the  legiti- 
mate methods  of  geometry  for  "  such  unsatisfactory  stuff  (as  I 
have  suggested  that  Barrow  would  consider  it)  as  analysis  "  ? 
Who  can  say  the  form  these  lectures  might  have  taken 
if  there  had  been  no  Newton ;  or  if  Barrow  had  taken 
kindly  to  Cartesian  geometry ;  or  what  a  second  edition, 
"revised  and  enlarged,"  might  have  contained,  if  Barrow 
on  his  return  to  Cambridge  as  Master  of  Trinity  and  Vice- 
Chancellor  had  had  the  energy  or  the  inclination  to  have 
made  one ;  or  if  Newton  had  made  a  treatise  of  it,  instead 
of  a  reprint  of  "  Scholastic  Lectures/'  as  Barrow  warns  his 
readers  that  it  is,  and  such  as  the  edition  of  1674  in  two  vol- 
umes probably  was  ?  But  Barrow  died  only  a  few  years  later, 
Newton  was  far  too  occupied  with  other  matters,  and  Collins 
seems  to  have  passed  out  of  the  picture,  even  if  he  had  been  the 
equal  of  the  other  two. 

VII 

DESCRIPTION    OF   THE   BOOK    FROM   WHICH 
THE   TRANSLATION    HAS   BEEN    MADE 

The  running  commentary  which  follows  is  a  precis  of 
a  full  translation  of  a  book  in  the  Cambridge  Library.  In 
one  volume,  bound  in  strong  yellow  calf,  are  the  two  works, 


INTR  OD  UCTION  2 1 

the  Lectiones  Opticce,  et  Geometricce ;  the  title-page  of  the 
first  bears  the  date  MDCLXIX,  that  of  the  second  the 
date  MDCLXX,  whilst  "  Imprimatur "  was  granted  on 
22nd  March  1669;  this  points  to  its  being  one  of  the 
original  combined  editions,  No.  4  of  the  list  in  Section  III 
of  this  Introduction.  On  the  title-page  of  the  Optics  there 
is  a  line  which  reads,  "To  which  are  annexed  a  few 
geometrical  lectures,"  agreeing  with  the  remark  in  the 
preface  to  the  geometrical  section  that  originally  there  were 
only  seven  geometrical  lectures  that  were  intended  to  be 
published  as  supplements  of  the  Optics,  instead  of  the 
thirteen  of  which  the  section  is  composed.  For  in  all 
probability  this  title-page  is  that  of  the  first  edition  of  the 
Optics,  but  the  Librarius,  whoever  he  may  have  been, 
persuaded  Barrow  to  leave  the  seven  lectures  out,  enlarge 
them  to  form  a  separate  work,  and  to  publish  them  as 
such  in  combination  with  the  Optics,  as  we  see,  in  1670; 
and  by  an  oversight  the  title-page  remained  uncorrected. 

Of  prefaces  there  are  three,  one  being  more  properly  an 
introduction,  explaining  the  plan  and  scope  of  the  originally 
designed  " XVIII  Lectures  on  Optics"  and  the  supple- 
mentary se'ven  geometrical  lectures ;  this  is  in  the  same  type 
as,  and  immediately  in  front  of,  the  Optics.  The  other  two 
are  true  prefaces  or  "Letters  to  the  reader";  they  are  in 
italics  :  a  full  translation  of  both  is  given  later. 

On  a  fly-leaf  in  front  of  the  Optics  is  a  list  of  symbols 
of  abbreviation  as  used  by  Barrow ;  as  these  cover  the  two 
sections  and  are  not  repeated  in  front  of  the  geometrical 
section,  they  furnish  additional  evidence  that  the  book  I 
have  used  is  one  of  the  first  combined  editions.  The 
similarity  of  the  symbols  used  by  Barrow  to  those  used 
at  the  present  day,  to  stand  for  quite  different  things^  does 
not  simplify  the  task  of  a  modern  reader.  This  is  especially 
the  case  with  the  signs  for  " greater  than"  and  "less  than/' 
where  the  "  openings  "  of  the  signs  face  the  reverse  way  to 
that  which  is  now  usual ;  another  point  which  might  lead 
to  error  by  a  casual  reader  who  had  not  happened  to 
notice  the  list  of  abbreviations,  is  the  use  of  the  plus  sign 
between  two  ratios  to  stand  for  the  ratio  compounded  from 
them,  i.e.  for  multiplication ;  the  minus  sign  does  not, 
however,  stand  for  the  ratio  of  two  ratios,  i.e.  for  division, 


22    BARROW'S   GEOMETRICAL  LECTURES 

the  ease  with  which  the  argument  may  be  followed  is  also  not 
by  any  means  increased  by  Barrow's  plan  of  running  his  work 
on  in  one  continuous  stream  (paper  was  dear  in  those  days), 
with  intermediate  steps  in  brackets;  and  this  is  made  still  worse 
by  the  use  of  the  "full  stop  "  as  a  sign  of  a  ratio  (division) 
instead  of  as  a  sign  of  a  rectangle  (multiplication)  ;  thus 
DH  .  HO  : :  (DL.  LN  : :  DL  -  DH  .  LN  -  HO  : :  LH  .  LB  : :  )LH  .  HK 
stands,  in  modern  symbols,  for  the  extended  statement 

DH-.HO  -  DL:LN  *  (DL- DH) :  (LN  -  HO)  -  LH:HB 

..-.     DH:HO  =  LH:LB  =  LH:HK; 

whereas  DL  x  LK-LH  x  HK=  KO  x  LH  -  HK  x  LH,  on  the 
contrary,  means,  as  is  usual  at  present,  DL.HK-LH.HK  = 
KO  .  LH  -  HK .  LH,  the  minus  sign  thus  being  a  weaker  bond 
than  that  of  multiplication,  but  a  stronger  bond  than  that 
of  ratio  or  division.  Barrow's  list  of  symbols,  in  full,  is : — 

"For  the  sake  of  brevity  certain   signs  are    used,    the 
meaning  of  which  is  here  subjoined. 
A  +  B  that  is,    A  and  B  taken  together. 

A  -  B  A,  B  being  taken  away. 

A  -  :  B  The  difference  of  A  and  B. 

A  x  B  A  multiplied  by,  or  led  into,  B. 

A  divided  by  B,  or  applied  to  B- 
B 

A  =  B  A  is  equal  to  B. 

Ac~B  A  is  greater  than  B. 

A-^B  A  is  less  than  B. 

A.  B  : :  C.  D  A  bears  to  B  the  same  ratio  as  C  to  D. 

A,  B,  C,  D  -H-  A,  B,  C,  D  are  in  continued  proportion. 

A .  B  c^C .  D  A  to  B  is  greater  than  C  to  D. 

A  to  B  is  less  than  C  to  D. 
= )  The  ratios       |       j  equal  to        | 

C-   M.N  AtoB;CtaOiareVgreaterthan|»M  to  N. 
— D)  compounded  (       )  less  than       ) 

The  square  on  A. 
The  side,  or  square  root  of,  A. 
The  cube  of  A. 
B-7  The  side  of  the  square  made  up  of  the 

square  of  A  and  the  square  of  B. 
Other  abbreviations,  if  there  are  any,  the  reader  will  re- 
cognise, by  easy  conjecture,  especially  as  I  have  used  very 
little  analysis." 


INTRODUCTION 


The  style  of  the  text,  as  one  would  expect  from  a  Barrow, 
is  "  classical " ;  that  is,  full  of  long  involved  sentences, 
phrases  such  as  "through  all  of  a  straight  line  points," 
general  inversion  of  order  to  enable  the  sense  to  run  on, 
use  of  the  relative  instead. of  the  demonstrative,  and  so  on; 
all  agreeing  with  what  is  but  an  indistinct  memory  (thank 
goodness  !)  of  my  trials  and  troubles  as  a  boy  over  Cicero, 
De  Senectute,  De  Amicitia,  and  such-like,  studied  (?),  by  the 
way,  in  Newton's  old  school  at  Grantham  in  Lincolnshire. 

In  this  way  there  is  a  striking  difference  between  the 
style  of  Barrow  and  the  straightforward  Latin  of  Newton's 
Prindpia,  as  it  stands  in  my  Latin  edition  of  1822,  by  Le 
Seur  and  Jacquier.  My  classical  attainments  are,  however, 
so  slight  that,  in  looking  for  possible  additions  by  Newton, 
I  have  preferred  to  rely  on  my  proof-reading  experience  in 
the  matter  of  punctuation.  The  strong  point  in  Barrow's 
somewhat  awful  punctuation  is  the  use  of  the  semicolon, 
combined  with  the  long  involved  sentence,  and  the  frequent 
interpolation  of  arguments,  sometimes  running  to  a  dozen 
lines,  in  parentheses ;  Newton  makes  use  of  the  short  con- 
cise sentence,  and  rarely  uses  the  semicolon,  nor  indeed 
does  he  use  the  colon  to  any  great  extent.  Of  course  I  do 
not  know  how  much  the  printer  had  to  do  with  the  punctua- 
tion in  those  days,  but  imagine  this  distinction  was  a  very 
great  matter  of  the  author.  Comparing  two  analogous 
passages,  from  each  author,  of  about  200-250  words,  we 
get  the  following  table  : — 


Barrow. 

Newton. 

II 

15 

commas 

10 

5 

semicolons 

None 

3 

colons 

5 

10 

full-stops 

4 

None 

parentheses 

This  contrast  is  striking  enough  for  all  practical  purposes  ; 
in  addition,  Barrow  starts  three  of  his  five  sentences  with  a 
relative,  whilst  Newton  does  not  do  this  once  in  his  ten. 

Using  this  idea,  I  failed  to  find  anything  that  could,  with 
any  probability,  be  ascribed  to  Newton, 


24   BARROWS  GEOMETRICAL  LECTURES 

Lastly,  one  strong  feature  in  the  book  is  the  continued  use 
of  the  paraboliforms  as  auxiliary  curves ;  this  corroborates 
my  contention  that  Barrow  fully  appreciated  the  importance 
and  inner  meaning  of  his  theorem,  or  rather  construction 
(see  note  to  Lect.  IX,  §  4) ;  that  is,  he  uses  it  in  precisely 
the  same  way  as  the  analytical  mathematician  uses  its  equi- 
valent, the  approximation  to  the  binomial  and  the  differentia- 
tion of  a  fractional  power  of  a  variable,  as  a  foundation  of 
all  his  work. 

Although  there  are  two  fairly  long  lists  of  errata,  most 
probably  due  to  Newton,  there  are  still  a  great  number  of 
misprints ;  the  diagrams  are,  however,  uniformly  good,  there 
being  no  omissions  of  important  letters  and  only  one  or  two 
slips  in  the  whole  set  of  200,  one  of  these  evidently  being 
the  fault  of  the  engraver;  nevertheless  they  might  have 
been  much  clearer  if  Barrow  had  not  been  in  the  habit 
of  using  one  diagram  for  a  whole  batch  of  allied  theorems, 
thereby  having  to  make  the  diagram  rather  complicated 
.  in  order  to  get  all  the  curves  and  lines  necessary  for  the 
whole  batch  of  theorems  on  the  one  figure,  whilst  only 
using  some  of  them  for  each  separate  theorem.  In  the 
text  which  follows  this  introduction,  only  those  figures  have 
been  retained  that  were  absolutely  essential. 

There  is  a  book-plate  bearing  a  medallion  of  George  I 
and  the  words  "  MUNIFICENTIA  REGIA  1715"  which  points 
out  that  the  book  I  have  was  one  of  the  30,000  volumes  of 
books  and  manuscripts  comprised  in  the  library  of  Bishop 
Moore  of  Norwich,  which  was  presented  to  the  Cambridge 
University  Library  in  1715  by  George  I,  as  an  acknowledg- 
ment of  a  loyal  address  sent  up  by  the  University  to  the  king 
on  his  accession.  It  may  have  come  into  his  possession  as 
a  personal  gift  from  Barrow  ;  at  any  rate,  there  is  an  in- 
scription on  the  first  fly-leaf,  "A  gift  from  the  author."  I 
am  unable  to  ascertain  whether  Moore  was  a  student  at 
Cambridge  at  the  date  of  the  publication  of  these  lectures, 
but  the  date  of  his  birth  (1646)  would  have  made  him  twenty- 
four  years  of  age  at  the  time,  and  this  supposition  would  ex- 
plain the  presence  of  a  four-line  Latin  verse  (Barrow  had  a 
weakness  for  turning  things  into  Latin  or  Greek  verse)  on 
the  back  of  the  title-page  of  the  geometrical  section,  which 
reads : — 


INTR  OD  UCTION  2  5 

To  a  young  man  at  the   University* 

Humble  work  of  thy  brother,  pronounced  or  to  be, 
Noiv  rightly  appears,  devoted  to  thee  ; 
Should 'st  learn  from  it  aught,  both  happy  and  sure 
In  thy  patronly  favour  permit  it  endure, 

and  is  in  the  same  handwriting  as  the  inscription. 


VIII 
THE   PREFACES 

In  the  following  translation  of  the  Prefaces,  ordinary  type 
is  used  instead  of  Barrow's  italics,  in  order  that  I  may  call 
attention  to  points  already  made,  or  points  that  will  be 
possibly  referred  to  later,  in  the  notes  on  the  text,  by  means 
of  italics. 

The  first  Preface,  which  precedes  the  Optics  :— 

"  Communication  to  the  reader. 

"  Worthy  reader, 

"  That  this,  of  whatever  humble  service  it 
may  be,  was  not  designed  for  you,  you  will  soon  understand 
from  many  indications,  if  you  will  only  deign  to  examine  it ; 
nor,  that  you  might  yet  demand  it  as  your  due,  were  other 
authorities  absent.  To  these  at  least,  truly  quaking  in  mind 
and  after  great  hesitation,  I  yielded  ;  chiefly  because  thereby 
I  should  set  as  an  example  to  my  successors  the  production 
of  a  literary  work  as  a  duty,  such  as  I  myself  was  the  first 
to  discharge ;  if  less  by  the  execution  thereof,  at  any  rate 
by  the  endeavour  at  advance,  not  unseemly,  nor  did  it  seem 
to  be  an  ostentation  foreign,  to  my  office.  There  was  in 
addition  some  slight  hope  that  there  might  be  therein  some- 
thing of  the  nature  of  good  fruit,  such  as  in  some  measure 
might  profit  you,  and  not  altogether  be  displeasing  to  you. 
Also,  remember,  I  warn  those  of  you,  who  are  more  ad- 
vanced in  the  subject  of  my  book,  what  manner  of  writing 
you  are  handling ;  not  elaborated  in  any  way  for  you  alone ; 

*  With  apologies  for  doggerel ;  but  the  translation  is  fairly  close,  line  for 
line. 


26    BARROW'S   GEOMETRICAL   LECTURES 

not  produced  on  my  own  initiative;  nor  by  long  medita- 
tion, exhibiting  the  ordered  concepts  of  leisurely  thought ; 
but  Scholastic   Lectures  ;   first  extracted  from  me  by  the 
necessities  of  my  office ;   then  from  time  to  time  expanded 
over-hastily  to  complete  my  task  within  the  allotted  time; 
lastly,  prepared  for  the  instruction  of  a  promiscuous  literary 
public,  for  whom  it  was  important  not  to  leave  out  many 
lighter  matters  (as  they  will  appear  to  you).     In  this  way 
you  will  not  be  looking  in  vain  (and  it  is  necessary  to  warn 
you  of  this,  lest  by  expecting  too  much  you  may  harm  both 
yourself  and  me)  for  anything  elaborated,  skilfully  arranged, 
or  neatly  set  in  order.     For  indeed  I  know  that,  to  make 
the  matter  satisfactory  to  you,  it  would  be  expedient  to  cut 
out  many  things,  to  substitute  many  things,  to  transpose 
many  things,  and  to  'recall  all  to  the  anvil  and  file.'     For 
this,  however,  I  had  neither  the  stomach  nor  the  leisure  to 
take  the  pains;  nor  indeed  had  I  the  capability  to  carry  the 
matter  through.     And  so  I  chose  rather  to  send  them  forth 
'  in  Nature's  garb,'  as  they  say,  and  just  as  they  were  born  ; 
rather  than,  by  laboriously  licking  them  into  another  shape, 
to   fashion  them  to  please.      However,  after   that    I    had 
entered  on  the  intention  of  publication,  either  seized  with 
disgust,  or  avoiding  the  trouble  to  be  undergone  in  making 
the  necessary  alterations,  in  order  that  I  should  not  indeed 
put  off  the  rewriting  of  the  greater  part  of  these  things,  as 
delicate  mothers  are  wont,  I  committed  to  the  foster  care  of 
friends,  not  unwillingly,  my  discarded  child,  to  be  led  out  and 
set  forth  as  it  might  seem  good  to  them.     Of  which,  for  I 
think  it  right  that  you  should  know  them  by  name,  Isaac 
Newton,  a  fellow  of  our  college  (a  man  of  exceptional  ability 
and  remarkable  skill)  has  revised  the  copy,  warning  me  of 
many  things  to  be  corrected,  and  adding  some  things  from 
his  own  work,  which  you  will  see  annexed  with  praise  here 
and  there.     The  other  (whom  not  undeservedly  I  will  call 
the  Mersenne  of  our  race,  born  to  carry  through  such  essays 
as  this,  both  of  his  own  work  and  that  of  others)  John 
Collins  has  attended  to  the  publication,  at  much  trouble 
to  himself. 

"  I  could  now  place  other  obstacles  to  your  expectation,  or 
show  further  causes  for  your  indulgence  (such  as  my  meagre 
ability,  a  lack  of  experiments,  other  cares  intervening)  if 


INTRODUCTION  27 

I  were  not  afraid  that  that  bit  of  wit  of  the  elder  Cato  would 
be  hurled  at  me  :— 

" '  Truly  you  publish  abroad  these  things  as  if  bound  by 
a  decree  of  the  Amphictyones.' 

"At  least  fairness  demanded  a  prologue  of  this  kind,  and 
in  some  degree  a  certain  parental  affection  for  one's  own 
offspring  enticed  it  forth,  in  order  that  it  might  stand  forth 
the  more  excusable,  and  more  defended  from  censure. 

"  But  if  you  are  severe,  and  will  not  admit  these  excuses 
into  a  propitious  ear,  according  to  your  inclination  (I  do 
not  mind)  you  may  reprove  as  much  and  as  vigorously  as 
you  please." 

The  second  Preface,  which  refers  to  the  Geometry : — 

"My  dear  reader, 

"  Of  these  lectures  (which you  will  now 

receive  tn  a  certain  measure  late-born),  seven  (one  being  ex- 
cepted)  I  intended  as  the  final  accompaniments  and  as  it  were 
the  things  left  over  from  the  Optical  lectures,  which  stand  forth 
lately  published;  otherwise,  I  imagine,  I  shall  be  thought 
little  of  for  bringing  out  sweepings  of  this  kind.  However, 
when  the  publisher  [or  editor — Librarius — ?  Collins]  thought, 
for  reasons  of  his  own,  that  these  matters  should  be  prepared, 
separately  removed  from  the  others ;  and  moreover  he  de- 
sired something  else  to  be  furnished  that  should  give  the 
work  a  distinct  quality  of  its  own  (so  that  indeed  it  might 
surpass  the  size  of  a  supplementary  pamphlet);  falling  in 
with  his  wishes  (/  will  not  say  very  willingly)  I  added  the 
first  five  lectures,  cognate  in  matter  with  those  following  and 
as  it  were  coherent ;  which  indeed  /  had  devised  some  years 
ago,  but,  as  with  no  idea  of  publishing,  so  without  that  care 
which  such  an  intention  calls  for.  For  they  are  clumsily 
and  confusedly  written;  nor  do  they  contain  anything 
firmly,  or  anything  lying  beyond  the  use  or  the  compre- 
hension of  the  beginners  for  which  they  are  adapted ;  where- 
fore I  warn  those  experienced  in  this  subject  to  keep  their 
eyes  turned  away  from  these  sections,  or  at  least  to  give 
them  indulgence  a  little  liberally. 

"  The  other  seven,  that  I  spoke  of,  I  expose  more  freely  to 
vour  view,  hoping  that  there  is  nothing  in  them  that  it  will 
displease  the  more  erudite  to  see. 


28   BARROW'S   GEOMETRICAL    LECTURES 

"  The  last  lecture  of  all  a  friend  (truly  an  excellent  man, 
one  of  the  very  best,  but  in  a  business  of  this  sort  an  in- 
satiable dun  *)  extorted  from  me ;  or,  more  correctly, 
claimed  its  insertion  as  a  right  that  was  deserved. 

"  For  the  rest,  what  these  lectures  bring  forth,  or  to  ivhat 
they  may  lead,  you  may  easily  learn  by  tasting  the  beginnings 
of  each. 

"  Since  there  is  now  no  reason  why  I  should  longer 
detain  or  delay  you, 

"FAREWELL." 


IX 
HOW   BARROW    MADE    HIS    CONSTRUCTIONS 

In  hazarding  a  guess  as  to  how  Barrow  came  by  his  con- 
structions, one  has,  to  a  great  extent,  to  be  guided  by  his 
other  works,  together  with  any  hint  that  may  be  obtained 
from  the  order  of  his  theorems  in  the  text.  Taking  the 
latter  first,  I  will  state  the  effect  the  reading  of  the  text  had 
on  me.  The  only  thing  noticeable,  to  begin  with,  was  the 
pairing  of  the  propositions,  rectangular  and  polar ;  the  rest 
seemed  more  or  less  a  haphazard  grouping,  in  which  one 
proposition  did  occasionally  lead  to  another;  but  certain 
of  the  more  difficult  constructions  were  apparently  without 
any  hint  from  the  preceding  propositions.  Once,  however, 
it  began  to  dawn  on  me  that  Barrow  was  trying  to  write  a 
complete  elementary  treatise  on  the  calculus,  the  matter  was 
set  in  a  new  light.  First,  the  preparation  for  the  idea  of  a 
small  part  of  the  tangent  being  substituted  for  a  small  part 
of  the  arc,  and  vice  versa  (Lect.  V,  §  6),  this,  of  course, 
having  been  added  later,  probably,  I  suggest,  to  put  the 
differential  triangle  on  a  sound  basis ;  then  the  lemmas  on 
hyperbolas,  for  the  equivalent  of  a  first  approximation  in 
the  form  of  y  =  (ax  +  b}J(cx  +  d)  for  any  equation  in  the  form 
giving  y  as  an  explicit  function  of  x ;  this  first  gave  the 
clue  pointing  to  his  constructions  having  been  found  out 
analytically ;  then  the  work  on  arithmetical  and  geometrical 
means  leading  to  the  approximation  to  the  binomial  raised 

*  Flagitator  improbus  ;  a  specimen  of  Harrovian  humour. 


INTRODUCTION  29 

to  a  fractional  power  ;  lastly,  a  few  tentative  standard  forms  ; 
and  then  Lect.  IX,  with  the  differentiation  of  a  fractional 
power,  and  the  whole  design  is  clear  as  day.  Barrow  knows 
the  calculus  algebraically  and  is  setting  it  in  geometrical 
form  to  furnish  a  rigorous  demonstration.  From  this  point 
onwards,  truly  with  many  a  sidestep  as  something  especially 
pretty  strikes  him  as  he  goes,  he  proceeds  methodically  to 
accumulate  the  usual  collection  of  standard  forms  and 
standard  rules  for  their  completion  as  a  calculus.  If  one 
judges  from  this  alone,  there  is  no  other  possible  explana- 
tion of  the  plan  of  the  work. 

I  then  looked  round  for  some  hint  that  might  corroborate 
this  opinion,  and  I  found  it,  to  me  as  clear  as  daylight,  in 
his  lectures  on  the  explanation  of  the  method  of  Archimedes. 
In  these  I  am  convinced  Barrow  is  telling  the  story  of  his 
oivn  method,  as  well  as  stating  the  source  from  which  he  has 
derived  the  idea  of  such  a  procedure.  With  this  compare 
Newton's  anagram  and  Fermat's  discreet  statement  of  the 
manner  in  which  he  proved  that  any  prime  of  the  form 
4^  +  i  was  the  sum  of  two  squares.  Any  reader,  who  has 
been  led,  by  reading  this  statement,  into  trying  to  produce 
a  proof  of  this  theorem  for  himself,  will  agree  with  me  that 
Fermat  was  not  giving  very  much  of  his  method  away.  And 
so  it  was  with  all  these  mathematicians,  and  other  scientists 
as  well  ;  they  stated  their  results  freely  enough,  and  some- 
times gave  proofs,  but  generally  in  a  form  that  did  not  reveal 
their  own  particular  methods  of  arriving  at  them.  For 
instance,  take  the  construction  of  Lect.  IX,  §  10  ;  to  my 
mind  there  cannot  possibly  be  any~~3oubt  that  he  arrived 
at  it  analytically;  and  the  analytical  equivalent  of  it  as  it 
stands  is 

If  v=f(x),     and     Mz=  Ny  +  (M-N)(w*  +  c\ 

then  Mdzldx=\\dyldx  +  (M  -  H)m; 


given  the  capacity  for  doing  this  bit  of  differentiation,  the 
construction  given  would  be  easily  found  by  Barrow.  This 
construction  is  all  the  more  remarkable  because  the  proof 
given  is  unsound,  not  to  say  wrong  ;  and  I  suggest  that  this 
fact  is  a  very  strong  piece  of  evidence  that  the  construction 
was  not  arrived  at  geometrically.  Many  other  examples 
might  be  cited,  but  this  one  should  be  sufficient. 


30   BARROW'S  GEOMETRICAL  LECTURES 


X 

ANALYTICAL   EQUIVALENTS   TO    BARROW'S 
CHIEF   THEOREMS 

Fundamental  Theorem 

If  n  is  any  positive  rational  number,  integral  or  fractional, 
then  (i  +  .tf)"<i  +  M.X,  according  as  //<  i  ;  and  this  inequality 
tends  to  become  an  equality  when  x  tends  to  zero. 
"  [Proved  without  convergence  in  Lect.  VII,  §§  13-16.] 


Standard  Forms  for  Differentiation 

1.  If y  is  any  function  of  x,  and  z  =  kjy9 

then  dzjdx=  -  (k/y>2).dy/dx        .         .         .   Lect.  VI II,     9 

2.  If  y  is  a  function  of  x,  and  z  =y  +  C, 

then  dzjdx  =  dyjdx Lect.  VIII,  11 

3.  If y  is  any  function  of  A:,  and  z1  =y'2  —  a2, 
then  z.dzjdx=y  .dyjdx  \  or,  in  another  form, 
\fz=  x/(y  -  a2),  then  dzjdx  =  [  v/J(y*  -  a*)]dyldx 

Lect.  VIII,  13 

4.  If  s2  =y'2  +  a2,  then  z  .  dzjdx  =y .  dyjdx, 

^r  dzldx  =  [yl/J(y<2  +  a2)]dyldx     .    '  .   Lect.  VIII,  14 

5.  If  z2  =  a2  -y2,  then  z.dzjdx  =  -y.dyjdx, 

or  dzjdx  =  -  [>/s/(aa  -y^dyjdx  .  .   Lect.  VIII,  1 5 

6.  Ifjy  is  any  function  of  x,  and  z  =  a  +  /% 

tfizn  dz,jdx  =  b  .dyfdx Lect.  IX,    i 

7.  Ifjy  is  any  function  of  x,  and  zn  =  an~r  .yr, 

then  (i/z) .  dz/dx  =  («//•) .  (i/y) .  ^//^  .         .  Lect.  IX,  3 

8.  Special  case  :  d(x")Jdx  =  «  .  x'1-1  or  «  .  (yjx\ 
where  n  is  a  positive  rational        .        ..         .   Lect.  IX,  4 

9.  The  case  when  n  is  negative  is  to  be  deduced  from 
the  combination  of  Forms  i  and  8. 

10.  If  y  =  tanx,  then  dyjdx^se^x,  proved  as  Ex.  5  on 
the  "differential  triangle"  at  the  end  of  Lect.  X. 

ir.  It  is  to  be  noted  that  the  same  two  figures,  as  used 
for  tan  x,  can  be  used  to  obtain  the  differential  coefficients 
of  the  other  circular  functions. 


INTRODUCTION  31 

Laws  for  Differentiation 

LAW  i.  Sum  of  Two  Functions.  —  If  w=y  +  z, 
then  dw/dx  =  dy/dx  +  dzjdx  ....  Lect.  VIII,  5 

LAW  2.  Product  of  Two  Functions.  —  Ifw=yz, 
then  (  i  jw)  .  dw/dx  =  (  i  /y)  -  dy/dx  +  (  r  /z)  .  dzjdx  .  Lect.  I  X,  1  2 

LAW  3.  Quotient  of  Two   Functions.  —  If  w=y/z, 
then,  if  v=  i/z,  (ijv)  .  dv/dx  =  -  (i/z)  .  dz/dx,  as  has  already 
been  obtained  in  Lect.  VIII,  9  ;  hence  by  the  above— 
(i/w)  .  dw/dx  =  (i/y)  .  dy/dx  -  (i/z)  .  dz/dx. 

N.B.  —  Note  the  logarithmic  form  of  these  two  results, 
corresponding  with  the  subtangents  used  by  Barrow. 

The   remaining    standard    forms    Barrow    is    unable   ap- 
parently to  obtain  directly;  and  the  same  remark  applies 
to  the  rest  of  the  laws.     So  he  proceeds  to  show  that 
Differentiation  and  Integration  are  inverse  operations. 
(i.)  If  R.z  =  [ydx,  then  R.dz/dx=y     .         .  Lect.  IX,  n 
(ii.)  If  R.  dz/dx  =y,  then  R.z  =  fydx     .         .  Lect.  XI,  19 
Hence   the    standard    forms   for    integration    are    to    be 
obtained  immediately  from  those  already  found  for  differ- 
entiation.    Barrow,  however,  proves  the  integration  formula 
for  an  integral  power  independently,  in  the  course  of  certain 
theorems  in  Lect.  XL    He  also  gives  a  separate  proof  of  the 
quotient  law  in  the  form  of  an  integration,  in  Lect.  XI,  27. 

Further  Standard  Forms  for  Integration 

A. 


:-.         .   Lect.    XII,   App.   3, 
B.   \{fi*dx  =  k(a*  -  i)  )  prob.  3,  4 


C.  (tanOdO  =  log(<;osO)          .         .   Lect.  XII,  App.  I,  2 

D.  \\sec  QdQ  =  \log{(i+  sin  Q)l(i-  sin  Q)}      „  „       5 

E.  fal(a?  -  x2)  =  {log  (a  +  x)/(a  -  x)}/2a    (see  Form  D) 

F.  fas  6  d(tan  6)  dO  =  \tan  0  d(cos  6)  dO  -  tan  0  cos  0, 
both  being   equal  to  ^secOdO,  the  only  example  of 
"integration  by  parts"  I  have  noticed  .  Lect.  XII,  App.  I,  8 

G.  dx/J(x*  +  ai)  =  /og[{x  +  J(x*  +  a*)}/a]    „  „        9 


32    BARROW'S   GEOMETRICAL  LECTURES 

Graphical  Integration  of  any  Function 

For  any  function,/^),  that  cannot  be  integrated  by  the 
foregoing  rules,  Barrow  gives  a  graphical  method  for 
^f(x)dx  as  a  logarithm  of  the  quotient  of  two  radii  vectores 
of  the  curve  r=f(0),  and  for  \dxjf(x)  as  a  difference  of 
their  reciprocals  .  .  .  Lect.  XII,  App.  Ill,  5-8 

Fundamental  Theorem  in  Rectification 

He  proves  that  (ds/dxf=  i  +  (dyjdxf  .  .  Lect.  X,  5 
He  rectifies  the  cycloid  (thus  apparently  anticipating 
Wren),  the  logarithmic  spiral,  and  the  three-cusped  hypo- 
cycloid  (as  special  cases  of  one  of  his  general  theorems),  and 
reduces  the  rectification  of  the  parabola  to  the  quadrature 
of  the  rectangular  hyberbola,  from  which  the  rectification 
follows  at  once. 

(XII,  App.  Ill,  i,  Ex.  2;  XI,  26;  XII,  20,  Ex.  3.) 

In  addition  to  the  foregoing  theorems  in  the  Infinitesimal 
Calculus  (for  if  it  is  not  a  treatise  on  the  elements  of  the 
Calculus,  what  is  it?),  Barrow  gives  the  following  interesting' 
theorems  in  the  appendix  to  Lect.  XI. : — 

Maxima  and  Minima 

He  obtains  the  maximum  value  of  xr{c—x)*t  giving  the 
condition  that  x/r  =  (c-x)/s;  also  he  shows  that  this  is  the 
condition  for  the  minimum  value  of  xr/(x  —  c)\ 

Trigonometrical  Approximations 

Barrow  proves  that  the  circular  measure  of  an  angle  a 
lies  between  3  sin  a/ (2  +  cos  a)  and  sin  a(2  +  cos  a)/(  i  +  2  cos  a), 
the  former  being  a  lower  limit,  and  equivalent  to  the  formula 
of  Snellius;  each  of  these  approximations  has  an  error  of 
the  order  of  a5. 


THE 
GEOMETRICAL  LECTURES 


ABRIDGED   TRANSLATION 

WITH  NOTES,  DEDUCTIONS,  PROOFS  OMITTED 

BY    BARROW,    AND    FURTHER    EXAMPLES    OF 

HIS   METHOD 


LECTURE    I 

Generation  of  magnitudes.  Modes  of  motion  and  the 
quantity  of  the  motive  force.  Time  as  the  independent  vari- 
able. Time,  as  an  aggregate  of  instants,  compared  with  a 
line,  as  the  aggregate  of  points.  Deductions. 

[In  this  lecture,  Barrow  starts  his  subject  with  what  he 
calls  the  generation  of  magnitudes.] 

Every  magnitude  can  be  either  supposed  to  be  produced, 
or  in  reality  can  be  produced,  in  innumerable  ways.  The 
most  important  method  is  that  of  "local  movements."  In 
motion,  the  matters  chiefly  to  be  considered  are  the  mode 
of  motion  and  the  quantity  of  the  motive  force.  Since 
quantity  of  motion  cannot  be  discerned  without  Time,  it 
is  necessary  first  to  discuss  Time.  /Time  denotes  not  an 
actual  existence,  but  a  certain  capacity  or  possibility  for  a 
continuity  of  existence;  just  as  space  denotes  a  capacity 
for  intervening  length.  Time  does  not  imply  motion,  as 
far  as  its  absolute  and  intrinsic  nature  is  concerned ;  not 
any  more  than  it  implies  rest ;  whether  things  move  or  are 
still,  whether  we  sleep  or  wake,  Time  pursues  the  even 
tenor  of  its  way.  Time  implies  motion  to  be  measurable ; 
without  motion  we  could  not  perceive  the  passage  of  Time.  / 


36  BARROW'S  GEOMETRICAL  LECTURES 

"  On  Time,  as  Time,  'tis  yet  confessed 
From  moving  things  distinct,  or  tranquil  rest, 
No  thought  can  be" 

is  not  a  bad  saying  of  Lucretius.  Also  Aristotle  says:  — 
"  When  we,  of  ourselves,  in  no  way  alter  the  train  of  our 
thought,  or  indeed  if  we  fail  to  notice  things  that  are  affecting 
it,  time  does  not  seem  to  us  to  have  passed"  And  indeed  it 
does  not  appear  that  any,  nor  is  it  apparent  how  much,  time 
has  elapsed,  when  we  awake  from  sleep.  But  from  this,  it 
is  not  right  to  conclude  that: — "//  is  plain  that  Time  does 
not  exist  without  motion  and  change  of  position"  "  We  do 
not  perceive  it,  therefore  it  does  not  exist,"  is  a  fallacious 
inference ;  and  sleep  is  deceptive,  in  that  it  made  us  connect 
two  widely  separated  instants  of  time.  However,  it  is  very 
true  that: — "  Whatever  the  amount  of  the  motion  was,  so 
much  time  seems  to  have  passed" ;  nor,  when  we  speak  of  so 
much  time,  do  we  mean  anything  else  than  that  so  much 
motion  could  have  gone  on  in  between,  and  we  imagine 
the  continuity  of  things  to  have  coextended  with  its  con- 
tinuously successive  extension. 

/  We  evidently  must  regard  Time  as  passing  with  a  steady 
flow;  therefore  it  must  be  compared  with  some  handy 
steady  motion,  such  as  the  motion  of  the  stars,  and 
especially  of  the  Sun  and  the  Moon ;  such  a  comparison 
is  generally  accepted,  and  was  born  adapted  for  the  pur- 
pose by  the  Divine  design  of  God  (Genesis  i,  14).  J  j  But  how, 
you  say,  do  we  know  that  the  Sun  is  carried  by  an  equal 
motion,  and  that  one  day,  for  example,  or  one  year,  is 
exactly  equal  to  another,  or  of  equal  duration  ?  I  reply 


LECTURE  I  37 

that,  if  the  sun-dial  is  found  to  agree  with  motions  of  any 
kind  of  time-measuring  instrument,  designed  to  be  moved 
uniformly  by  successive  repetitions  of  its  own  peculiar 
motion,  under  suitable  conditions,  for  whole  periods  or  for 
proportional  parts  of  them ;  then  it  is  right  to  say  that  it 
registers  an  equable  motion. i  It  seems  to  follow  that  strictly 
speaking  the  celestial  bodies  are  not  the  first  and  original 
measures  of  Time;  but  rather  those  motions,  which  are 
observed  round  about  us  by  the  senses  and  which  underlie 
our  experiments,  since  we  judge  the  regularity  of  the 
celestial  motions  by  the  help  of  these.  On  the  other  hand, 
Time  may  be  used  as  a  measure  of  motion ;  just  as  we 
measure- space  from  some  magnitude,  and  then  use  this 
space  to  estimate  other  magnitudes  commensurable  with 
the  first  j  i.e.  we  compare  motions  with  one  another  by  the 
use  of  time  as  an  intermediary. 

Time  has  many  analogies  with  a  line,  either  straight  or 
circular,  and  therefore  may  be  conveniently  represented  by 
it ;  for  time  has  length  alone,  is  similar  in  all  its  parts,  and 
can  be  looked  upon  as  constituted  from  a  simple  addition 
of  successive  instants  or  as  from  a  continuous  flow  of  one 
instant ;  either  a  straight  or  a  circular  line  has  length  alone, 
is  similar  in  all  its  parts,  and  can  be  looked  upon  as  being 
made  up  of  an  infinite  number  of  points  or  as  the  trace  of 
a  moving  point. 

Quantity  of  the  motive  force  can  similarly  be  thought  of 
as  aggregated  from  indefinitely  small  parts,  and  similarly 
represented  by  a  straight  line  or  a  circular  line ;  when  Time 
is  represented  by  a  distance  the  motive  force -is  the  same 


38  BARROWS  GEOMETRICAL  LECTURES 

as  the  velocity.  Quantity  of  velocity  cannot  be  found  from 
the  quantity  of  the  space  traversed  only,  nor  from  the  lime 
taken  only,  but  from  both  of  these  brought  into  reckoning 
together ;  and  quantity  of  time  elapsed  is  not  determined 
without  known  quantities  of  space  and  velocity;  nor  is 
quantity  of  space  (so  far  as  it  may  be  found  by  this 
method)  dependent  on  a  definite  quantity  of  velocity 
alone,  nor  on  so  much  given  time  alone,  but  on  the  joint 
ratio  of  both. 

To  every  instant  of  time,  or  indefinitely  small  particle  of 
time,  (I  say  instant  or  indefinite  particle,  for  it  makes  no 
difference  whether  we  suppose  a  line  to  be  composed  of 
points  or  of  indefinitely  small  linelets ;  and  so  in  the  same 
manner,  whether  we  suppose  time  to  be  made  up  of  instants 
or  indefinitely  minute  timelets) ;  to  every  instant  of  time,  I 
say,  there  corresponds  some  degree  of  velocity,  which  the 
moving  body  is  considered  to  possess  at  the  instant ;  to  this 
degree  of  velocity  there  corresponds  some  length  of  space 
described  (for  here  the  moving  body  is  a  point,  and  so  we 
consider  the  space  as  merely  long).  But  since,  as  far  as 
this  matter  is  concerned,  instants  of  time  in  nowise  depend 
on  one  another,  it  is  possible  to  suppose  that  the  moving 
body  in  the  next  instant  admits  of  another  degree  of  velo- 
city (either  equal  to  the  first  or  differing  from  it  in  some 
proportion),  to  which  therefore  will  correspond  another 
length  of  space,  bearing  the  same  ratio  to  the  former  as  the 
latter  velocity  bears  to  the  preceding;  for  we  cannot  but 
suppose  that  our  instants  are  exactly  equal  to  one  another. 
Hence,  if  t.a  every  instant  of  time  there  is  assigned  a  suit- 


LECTURE  I  39 

able  degree  of  velocity,  there  will  be  aggregated  out  of  these 
a  certain  quantity,  to  any  parts  of  which  respective  parts 
of  space  traversed  will  be  truly  proportionate;  and  thus 
a  magnitude  representing  a  quantity  composed  of  these 
degrees  can  also  represent  the  space  described.  Hence, 
if  through  all  points  of  a  line  representing  time  are  drawn 
straight  lines  so  disposed  that  no  one  coincides  with  another 
(i.e.  parallel  lines),  the  plane  surface  that  results  as  the  aggre- 
gate of  the  parallel  straight  lines,  when  each  represents  the 
degree  of  velocity  corresponding  to  the  point  through  which 
it  is  drawn,  exactly  corresponds  to  the  aggregate  of  the 
degrees  of  velocity,  and  thus  most  conveniently  can  be 
adapted  to  represent  the  space  traversed  also.  Indeed 
this  surface,  for  the  sake  of  brevity,  will  in  future  be  called 
the  aggregate  of  the  velocity  or  the  representative  of  the 
space.  It  may  be  contended  that  rightly  to  represent  each 
separate  degree  of  velocity  retained  during  any  timelet,  a 
very  narrow  rectangle  ought  to  be  substituted  for  the  right 
line  and  applied  to  the  given  interval  of  time.  Quite  so, 
but  it  comes  to  the  same  thing  whichever  way  you  take  it\  but 
as  our  method  seems  to  be  simpler  and  clearer,  we  will  in 
future  adhere  to  it. 

[Barrow  then  ends  the  lecture  with  examples,  from  which 
he  obtains  the  properties  of  uniform  and  uniformly  accel- 
erated motion.] 

(i)  If  the  velocity  is  always  the  same,  it  is  quite  evident 
from  what  has  been  said  that  the  aggregate  of  the  velocity 
attained  in  any  definite  time  is  correctly  represented  by  a 
parallelogram,  such  as  AZZE,  where  the  side  AE  stands  for 


40  BARROW'S  GEOMETRICAL  LECTURES 


a  definite  time,  the  other  AZ,  and  all  the  parallels  to  it, 

BZ,  CZ,  DZ,  EZ,  separate  degrees  of 

velocity  corresponding  to  the  separate 

instants   of  time,   and   in   this   case 

plainly  equal  to  one  another.     Also 

the  parallelograms  AZZB,  AZZC,  AZZD, 

AZZE,  conveniently  represent,  as  has 


Fig.  i. 


been  said,  the  spaces  described  in  the  z 
respective  times,  AB,  AC,  AD,  AE. 

(2)  If  the  velocity  increase  uniformly  from  rest,  then  the 
aggregate  of  the  velocities  is  represented  by  the  triangle 
AEY.     Also  if  the  velocity  increases  uniformly  from  some 
definite  velocity  to  another   definite   velocity  represented 
respectively  by  CY,  EY,  then  the  space  is  represented  by  a 
trapezium,  such  as  CYYE. 

(3)  If  the  velocity  increase  according  to  a  progression  of 
square  numbers,  the  space  described  to  represent  the  aggre- 
gate of  the  velocity  is  the  complement  of  a  semi-parabola. 
[For  which  Barrow  gives  a  figure.] 

[From  (i)  and  (2)  all  the  properties  of  uniform  motion 
and  of  uniformly  accelerated  motion  are  simply  deduced, 
and  the  lecture  concludes  with  the  remark  : — ] 

These  things,  being  necessary  for  the  understanding  of 
things  to  be  said  later,  and  theories  of  motion  that  are,  I 
think,  not  on  the  whole  quite  useless,  it  has  seemed  to  be 
advantageous  to  explain  clearly  as  a  preliminary.  Having 
finished  this  task,  I  direct  my  steps  forward. 


LECTURE  I  41 


NOTE 

There  is  not  much  in  this  lecture  calling  for  remark. 
The  matter,  as  Barrow  says  in  his  Preface,  is  intended  for 
beginners.  There  is,  however,  the  point,  to  which  attention 
is  called  by  the  italics  on  page  39,  that  Barrow  fails  to  see 
any  difference  between  the  use  of  lines  and  narrow  rect- 
angles as  constituent  parts  of  an  area.  This  is  Cavalieri's 
method  of  "indivisibles,"  which  Pascal  showed  incontro- 
vertibly  was  the  same  as  the  method  of  "exhaustions," 
as  used  by  the  ancients.  There  is  evidence  in  later  lectures 
that  Barrow  recognized  this;  for  he  alludes  to  the  possi- 
bility of  an  alternative  indirect  argument  (discursus  apo- 
gogicus)  to  one  of  his  theorems,  and  later  still  shows  his 
meaning  to  be  the  method  of  obtaining  an  upper  and  a 
lower  limit.  There  is  also  a  suggestion  that  he  personally 
used  the  general  modern  method  of  the  text-books,  that  of 
proving  that  the  error  is  less  than  a  rectangle  of  which  one 
side  represented  an  instant  and  the  other  the  difference 
between  the  initial  and  final  velocities ;  and  that  it  could 
be  made  evanescent  by  taking  the  number  of  parts,  into 
which  the  whole  time  was  divided,  large  enough.  Also 
the  attention  of  those  who  still  fight  shy  of  graphical 
proofs  for  the  laws  of  uniformly  accelerated  motion,  if  any 
such  there  be  to-day,  is  called  to  the  fact  that  these  proofs 
were  given  by  such  a  stickler  for  rigour  as  Barrow,  with  the 
remark  that  they  are  evident,  at  a  glance,  from  the  diagrams 
he  draws. 


LECTURE    II 

Generation  of  magnitudes  by  "local  movements"  The 
simple  motions  of  translation  and  rotation. 

Mathematicians  are  not  limited  to  the  actual  manner  in 
which  a  magnitude  has  been  produced ;  they  assume  any 
method  of  generation  that  may  be  best  suited  to  their 
purpose.*  Magnitudes  may  be  generated  either  by  simple 
motions,  or  by  composition  of  motions,  or  by  concurrence 
of  motions. 

[Examples  of  the  difference  of  meaning  that  Barrow 
attaches  to  the  two  latter  phrases  are  given  by  him  in  a 
later  lecture.  The  simple  motions  are  considered  in  this 
lecture.] 

There  are  two  kinds  of  simple  motions,  translation  and 
rotation,  i.e.  progressive  motion  and  motion  in  a  circle, 
For  these  motions,  mathematicians  assume  that  (i)  a 
point  can  progress  straightforwardly  from  any  fixed 
terminus,  and  describe  a  straight  line  of  any  length ; 
(2)  a  straight  line  can  proceed  with  one  extremity  moving 


*  As  an  example,  take  the  case  of  finding  the  volume  of  a  right  circular 
cone  by  integration ;  here,  by  definition,  the  method  of  generation  is  the 
rotation  of  a  right-angled  triangle  about  •  one  of  the  rectangular  sides  ; 
but  it  is  supposed  to  be  generated,  for  the  purpose  of  modern  integration, 
by  the  motion  of  a  circle,  that  constantly  increases  in  size,  and  moves 
parallel  to  itself  with  its  centre  on  the  axis  of  the  cone. 


LECTURE  II  43 

along  any  other  line,  keeping  parallel  to  itself;  the  former 
is  called  the  genetrix^  and  is  said  to  be  applied  to  the  latter 
which  is  called  the  directrix ;  by  these  are  described  paral- 
lelogrammatic  surfaces,  when  the  genetrix  and  the  directrix 
are  both  in  the  same  plane,  and  prismatic  and  cylindrical 
surfaces  otherwise.  In  general,  the  genetrix  may,  if  neces- 
sary, be  taken  as  a  curve,  which  is  intended  to  include 
polygons,  and  the  genetrix  and  the  directrix  may  usually 
be  interchanged.  The  same  kinds  of  assumptions  are 
made  for  simple  motions  of  rotation ;  and  by  these  are 
described  circles  and  rings  and  sectors  or  parts  of  these, 
when  a  straight  line  rotates  in  its  own  plane  about  a  point 
in  itself  or  in  the  line  produced;  if  the  directrix  is  a  curve 
(in  the  wider  sense  given  above),  and  does  not  lie  in  the 
plane  of  the  genetrix,  of  which  one  point  is  supposed  to 
be  fixed,  the  surfaces  generated  are  pyramidal  or  conical. 
From  this  kind  of  generation  is  deduced  the  similarity  of 
parallel  sections  of  such  surfaces ;  and  thus  it  is  evident 
that  the  surfaces  can  also  be  generated  by  taking  the 
genetrix  of  the  first  method  as  the  directrix,  and  the  former 
directrix  as  the  genetrix  so  long  as  it  is  supposed  to  shrink 
proportionately  as  it  proceeds  parallel  to  itself  towards  what 
was  the  vertex  or  fixed  point  in  the  first  method. 

For  producing  solids  the  chief  method  is  a  simple 
rotation,  about  some  fixed  line  as  axis,  of  another  line 
lying  in  the  same  plane  with  it.  In  addition,  there  is  the 
method  of  "indivisibles,"  which  in  most  cases  is  perhaps 
the  most  expeditious  of  all,  and  not  the  least  certain  and 
infallible  of  the  whole  set. 


44  BARROWS  GEOMETRICAL  LECTURES 

The  learned  A.  Tacquetus  *  more  than  once  objects  to 
this  method  in  his  clever  little  book  on  "  Cylindrical  and 
Annular  Solids,"  and  therein  thinks  that  he  has  falsified 
it,  because  the  things  found  by  means  of  it  concerning  the 
surfaces  of  cones  and  spheres  (I  mean  quantities  of  these) 
do  not  agree  in  measurement  with  the  truths  discovered 
and  handed  down  by  Archimedes. 


Fig.  io.t 

Take,  for  example,  a  right  cone  DVY,  whose  axis  is  VK; 
through  every  point  of  this  suppose  that  there  pass  straight 
lines  ZA,  ZB,  ZC,  ZD  (or  KD),  etc.;  from  these  indeed 
according  to  the  Atomic  theory  the  right-angled  triangle 
VDK  is  made  up;  and  from  the  circles  described  with  these 
as  radii  the  cone  itself  is  made.  "  Therefore"  he  argues, 
" from  the  peripheries  of  these  circles  is  composed  the  conical 
surface  ;  now  this  is  found  to  be  contrary  to  the  truth  ;  hence 
the  method  is  fallacious" 

I  reply  that  the  calculation  is  wrongly  made  in  this 
manner ;  and  in  the  computation  of  the  peripheries  of  which 

*  Andreas  Tacquet,  a  Jesuit  of  Antwerp :  published  a  book  on  the 
cylinder  (1651),  Elementa  Geometrice  (1654)  and  a  book  on  Arithmetic 
(1664);  mentioned  by  Wallis. 

f  The  numbering  of  the  diagrams  is  Barrow's  and  is,  in  consequence  of 
abridgment,  not  consecutive. 


LECTURE  II  45 

such  a  surface  is  composed,  a  reasoning  has  to  be  adopted 
different  from  that  used  when  computing  the  lines  from 
which  plane  surfaces  are  made  up,  or  the  planes  from  which 
solids  are  formed.  In  fact,  it  must  be  considered  that  the 
multitude  of  peripheries  forming  the  curved  surface  are 
produced,  through  the  rotation  of  the  line  VD,  from  the 
multitude  of  points  in  the  genetrix  VD  itself ";  by  observing 
this  distinction  all  error  will  be  obviated,  as  I  will  now 
demonstrate.* 

At  every  point  of  the  line  VD,  instead  of  the  line  VK, 
suppose  that  right  lines  are  applied  perpendicular  to  the 
line  VD,  and  equal  to  the  peripheries,  taken  in  order, 
that  make  up  the  curved  surface.  From  these  parallel 
straight  lines  is  generated  the  plane  VDX,  which  is  equal  to 
the  said  curved  surface. 

Further,  if  instead  of  the  peripheries  we  apply  the  corre- 
sponding radii,  the  space  produced  will  bear  to  the  curved 
surface  a  ratio  equal  to  that  of  the  radius  of  any  circle  to 
its  circumference.  In  the  particular  case  chosen,  the  two 


*  This  is  the  first  example  we  come  across  of  the  superiority  of  Barrow's 
insight  into  what  is  really  the  method  of  integration.  In  effect,  Barrow 
points  out  that  if  a  periphery  is  thought  of  as  a  solid 
ring  of  very  minute  section,  then  in  this  case  the 
section  is  a  trapezium,  as  shown  in  the  annexed 
diagram,  of  which  the  parallel  sides  are  perpendicular 
to  the  axis  of  the  cone,  and  the  non-parallel  sides 
both  pass,  if  produced,  through  the  vertex.  Tacquet 
uses  the  surface  generated  by  the  top  parallel  sides, 
PS  as  if  he  were  finding  the  area  of  a  circle,  by 
means  of  concentric  rings  (?  or  he  uses  the  perpendi- 
cular distance  from  S  on  QR) ;  Barrow  points  out  that 
he  should  use  the  surface  generated  by  the  slant 
side  SR. 

In  modern  phraseology,  Barrow  shows  that  Tacquet 
has  made  the  error  of  integrating  along  a  radius  of  the 
base  (?  or  along  the  axis),  instead  of  along  a  slant  side. 


46  BARROW'S  GEOMETRICAL  LECTURES 

plane  surfaces  are  triangles  and  the  area   of  the  curved 
surface  is  thus  easily  found. 

There  are  other  methods  which  may  be  used  conveniently 
in  certain  cases ;  but  enough  has  been  said  for  the  present 
concerning  the  construction  of  magnitudes  by  simple 
motions. 


NOTE 

It  would  be  interesting  to  see  how  Barrow  would  get 
over  the  difficulty  raised  by  Tacquet,  if  Tacquet's  example 
had  been  the  case  of  the  oblique  circular  cone.  It  seems 
to  me  to  be  fortunate  for  Barrow  that  this  was  not  so. 
Barrow  also  states,  be  it  noted,  that  the  method  is  general 
for  any  solid  of  revolution,  if  the  generating  line  is  supposed 
to  be  straightened  before  the  peripheries  are  applied;  in 
which  case,  the  area  can  be  found  for  the  curved  surface 
only  when  the  plane  surface  aggregated  from  the  applied 
peripheries  turns  out  to  be  one  whose  dimensions  can  be 
found. 

Thus,  if  the  ordinate  varies  as  the  square  of  the  arc 
measured  from  the  vertex,  the  plane  equivalent  is  a  semi- 
parabola,  and  the  area  is  2^5^/3,  where  s  is  length  of  the 
rotating  arc,  and  r  is  the  maximum  or  end  ordinate, 


LECTURE    III 


Composite  and  concurrent  motions.  Composition  of 
rectilinear  and  parallel  motions. 

In  generation  by  composite  motions,  if  the  remaining 
motions  are  unaltered,  then,  according  as  the  velocity  of 
one,  or  more,  is  altered,  we  usually  obtain  magnitudes 
differing  not  only  in  kind  but  also  in  quantity,  or  at  least 
differing  in  position  every  time. 

Thus,  suppose  the  straight  line  AB      jyi A 

is  carried  along  the  straight  line  AC 
by  a  uniform  parallel  motion,  and  at 
the  same  time  a  point  M  descends 
uniformly  in  AB;  or  suppose  that, 
while  AC  descends  with  a  uniform 
parallel  motion,  it  cuts  AB  also 
moving  uniformly  and  to  the  right.  A 
From  motions  of  this  kind,  com- 
posite in  the  former  case,  and  con- 
current in  the  latter,  the  straight  line  AM  may  be 
produced. 

Again,  if  in  the  previous  example,  whilst  the  motion  of 
the  straight  line  AB  remains  the  same  as  before  with  respect 
to  its  velocity,  but  the  uniform  motion  of  the  point  M,  or 


48  BARROW'S  GEOMETRICAL  LECTURES 

of  the  straight  line  AC,  is  altered  in  velocity,  so  that  indeed 
the  point  M  now  comes  to  the  point  //,,  or  AC  cuts  AB  in 
/*,,  there  is  described  by  this  motion  another  straight  line 
A/*,  in  a  different  position  from  the  first. 

Further,  if  once  more,  while  the  motion  of  AB  remains 
the  same,  instead  of  the  uniform  motion  of  the  point  M, 
or  of  the  straight  line  AC,  we  substitute  a  motion  that  is 
called  uniformly  accelerated ;  from  such  composite  or  con- 
current motion  is  produced  the  parabolic  line  A  MX,  or  in 
another  case  the  line  A^Y,  according  as  the  accelerated 
motion  is  supposed  to  be  one  thing  or  another  in  degree. 

In  these  examples,  it  is  seen  that  composite  and  con- 
current motions  come  to  the  same  thing  in  the  end ;  but 
frequently  the  generation  of  magnitudes  is  not  so  easily 
to  be  exhibited  by  one  of  these  methods  as  by  the  other. 
Thus  suppose  that  a  straight  line  AB  is  uniformly  rotated 
round  A,  and  at  the  same  time  the  point  M,  starting  from  A, 
is  carried  along  AB  by  a  continuous  and  uniform  motion; 
from  this  composite  motion  is  produced  a  certain  line,  namely 
the  Spiral  of  Archimedes,  which  cannot  be  explained  satis- 
factorily by  any  concurrence  of  motions.  On  the  other  hand, 
if  a  straight  line  BA  is  rotated  with  uniform  motion  about  a 
centre  B,  and  at  the  same  time  a  straight  line  AC  is  moved 
in  a  pamllelwise  manner  uniformly  along  AB,  the  continuous 
intersections  of  BA,  AC,  so  moving,  form  a  certain  line, 
usually  called  the  Quadratrix ;  and  the  generation  of  this 
line  is  not  so  clearly  shown  or  explained  by  any  strictly 
so-called  composition  of  motion. 

Magnitudes  can  be  compounded  and  also  decomposed  in 


LECTURE  III  49 

innumerable  ways ;  but  it  is  impossible  to  take  account  of 
all  of  these,  so  we  shall  only  discuss  some  important  cases, 
such  as  are  considered  to  be  of  most  service  and  the  more 
easily  explained.  Such  especially  are  those  that  are  com- 
pounded of  rectilinear  and  parallel  motions,  or  rectilinear  and 
rotary  motions,  or  of  several  rotary  motions;  preference 
being  given  to  those  in  which  the  constituent  simple  motions 
are  all,  or  at  least  some  of  them,  uniform.  Moreover,  there 
is  not  any  magnitude  that  cannot  be  considered  to  have  been 
generated  by  rectilinear  motions  alone.  For  every  line  that 
lies  in  a  plane  can  be  generated  by  the  motion  of  a  straight 
line  parallel  to  itself,  and  the  motion  of  a  point  along  it ; 
every  surface  by  the  motion  of  a  plane  parallel  to  itself  and 
the  motion  of  a  line  in  it  (that  is,  any  line  on  a  curved  surface 
can  be  generated  by  rectilinear  motions) ;  in  the  same  way 
solids,  which  are  generated  by  surfaces,  can  be  made  to 
depend  on  rectilinear  motions. 

I  will  only  consider  the  generation  of  lines  lying  in  one 
plane  by  rectilinear  and  parallel  motions ;  for  indeed  there 
is  not  one  that  cannot  be  produced  by  the  parallel  motion 
of  a  straight  line  together  with  that  of  a  point  carried  along 
it ;  *  but  the  motions  must  be  combined  together  as  the 
special  nature  of  the  line  demands. 

For  instance,  suppose  that  a  straight  line  ZA  is  always 
moved  along  the  straight  line  AY  parallelwise,  by  any 
motion,  Uniform  or  variable,  increasing  or  decreasing 


*  In  other  words,  Barrow  states  that  ever}'  plane  curve  has  a  Cartesian 
equation,  referred  to  either  oblique  or  rectangular  coordinates ;  yet  it  is 
doubtful  whether  he  fully  recognizes  that  all  the  properties  of  the  curve  can 
be  obtained  from  the  equation. 


50  BARROW'S  GEOMETRICAL  LECTURES 


or  alternating  in  velocity,  according  to  any  imaginable 
ratio ;  and  that  any  point  M  in  it  is  moved  in  such  a 
way  that  the  motion  of  the  point  is  proportional  to  the 
motion  of  the  straight  line,  throughout  any  the  same 
intervals  of  time ;  then  there  will  be  certainly  a  straight 
line  generated  by  these  motions. 

A B       C 


\ 


2 
Fig.  17- 

For,  since  we  always  have 

AB:AC  =  BM:C/*,     or     AB  :  MX  =  AM  :  X/*, 
(MX  being  drawn  parallel  to  AC),  it  follows  that  the  points 
A,  M,  ft  are  in  one  straight  line. 

But  if,  instead,  these  motions  are  comparable  with  one 
another  in  such  fashion  that,  given  any  line  D,  then  the 
rectangle,  contained  by  the  difference  between  the  line  D 
and  BM  the  distance  traversed  and  BM  itself,  always  bears 
the  same  ratio  to  the  square  on  AB  (the  distance  traversed 
by  the  line  AZ  in  the  same  time) ;  then  a  circle  or  an  ellipse 
is  described ;  a  circle,  if  the  supposed  ratio  is  one  of  equality 
and  the  angle  ZAY  is  a  right  angle,  and  an  ellipse  otherwise  . 
and  in  these  there  will  be  one  diameter,  situated  in  the  line 
A2  in  its  first  position,  and  drawn  from  A  in  the  direction 
of  Z,  and  this  diameter  will  be  equal  to  D. 


LECTURE  III  51 

If,  however,  the  motions  are  such  that  the  rectangle  con- 
tained by  the  sum  of  the  lines  D  and  BM  and  BM  itself  bears 
always  the  same  ratio  to  the  square  on  the  line  AB,  a  hyper- 
bola will  be  produced;  a  rectangular  or  equilateral  hyperbola, 
if  the  assigned  ratio  is  one  of  equality  and  the  angle  ZAY  is 
a  right  angle;  if  otherwise,  of  another  kind,  according  to 
the  quality  of  the  assigned  ratio;  of  these  the  transverse 
diameter  will  be  equal  to  D,  being  situated  in  ZA  when 
occupying  its  first  position,  and  being  measured  in  a 
direction  opposite  to  Z;  and  the  parameter  is  given  by 
the  given  ratio. 

But  if  the  rectangle  contained  by  D  and  BM  bears  always 
the  same  ratio  to  the  square  on  AB,  it  is  evident  that  a 
parabolic  line  is  produced,  of  which  the  parameter  is  easily 
found  from  the  given  straight  line  D  and  the  quantity  of  the 
assigned  ratio. 

Also  in  the  first  case  of  these,  if  the  transverse  motion 
along  AY  is  supposed  to  be  uniform,  the  descending  motion 
along  AZ  will  also  be  uniform;  in  the  second  and  third 
cases,  if  the  motion  along  AY  is  uniform,  the  descending 
motion  along  AZ  will  be  continually  increasing;  and  the 
same  thing  being  supposed  in  the  last  case,  in  which  the 
parabola  is  produced,  the  point  M  has  its  velocity  increased 
uniformly. 

In  a  similar  manner,  any  other  line  can  be  conceived  to 
be  produced  by  such  a  composition  of  motion.  But  we 
shall  come  across  these  some  time  or  other  as  we  go  along ; 
let  us  see  whether  anything  useful  in  mathematics  can  be 
obtained  from  a  supposed  generation  of  lines  in  this  way. 


52  BARROWS  GEOMETRICAL  LECTURES 

But  for  the  sake  of  simplicity  and  clearness  let  us  suppose 
that  one  of  the  two  motions,  say  that  of  the  line  preserving 
parallelism,  is  always  uniform;  and  let  us  strive  to  make 
out  what  general  properties  of  the  generated  lines  arise  from 
the  general  differences  with  regard  to  the  velocity  of  the 
other ;  let  us  try,  I  say,  but  in  the  next  lecture. 

NOTE 

We  here  see  the  reason  for  Barrow  considering  time  as 
the  independent  variable ;  he  states,  indeed,  that  the  con- 
structions can  be  effected,  no  matter  what  is  the  motion  of 
the  line  preserving  parallelism  ;  but  for  the  sake  of  simplicity 
and  clearness  he  decides  to  take  this  motion  as  uniform ;  for 
this  the  consideration  of  time  is  necessary.  At  the  same 
time  it  is  to  be  noted  that  Barrow,  except  for  the  first  case 
of  the  straight  line,  is  unable  ^o^exnHcitlyy describe  the 
velocity  of  the  point  M,  and  uses  a  geometrical  condition 
as  the  law  of  the  locus ;  in  other  words,  he  gives  the  pure 
geometry  equivalent  of  the  Cartesian  equation.  In  later 
lectures,  we  shall  find  that  he  still  further  neglects  the  use 
of  time  as  the  independent  variable.  This,  as  has  been 
explained  already,  is  due  to  the  fact  that  the  first  five 
lectures  were  added  afterwards.  In  Barrow's  original  de- 
sign, the  independent  variable  is  a  length  along  one  of  his 
axes.  This  length  is,  it  is  true,  divided  into  equal  parts ; 
but  that  is  the  only  way,  a  subsidiary  one,  in  which  time 
enters  his  investigations ;  and  even  so,  the  modern  idea  of 
"  steps  "  along  a  line,  used  in  teaching  beginners,  is  a  better 
analogue  to  Barrow's  method  that  that  which  is  given  by  a 
comparison  with  fluxions. 


LECTURE    IV 

Properties  of  curves  arising  from  composition  of  motions. 
The  gradient  of  the  tangent.  Generalization  of  a  problem  of 
Galileo.  Case  of  infinite  velocity. 

Hereafter,  for  the  sake  of  brevity,  I  shall  call  a  parallel 
motion  of  the  straight  line  AZ  along  AY  a  "transverse 
motion,"  and  the  motion  of  a  point  moving  from  A  in  the 
line  AZ  a  "descent"  or  a  "descending  motion,"  regard 
being  had  of  course  to  the  given  figure.  Also  I  shall  take 
the  motion  along  AY  and  parallels  to  it  as  uniform,  hence 
this  motion  and  parts  of  it  can  represent  the  time  and 
parts  of  the  time.  Now  I  come  to  the  properties  of  lines 
produced  by  a  uniform  transverse  motion  and  a  continually 
increasing  descent. 

1.  The  line  produced  is  curved  in  all  its  parts. 

2.  The  velocity  of  the  uniform  descending  motion,  by 
which  a  curve  is  described  (i.e.  if  there  is  a  common  uniform 
transverse  motion  for  the  chord  and  its  arc)  is  less  than  the 
velocity,  which  the  increasing  descending  motion  has  at  N, 
the  common  end  of  both. 

3.  Of  a  curve  of  this  sort,  any  chord,  as  MO,  falls  entirely 


54  BARROW'S  GEOMETRICAL  LECTURES 

within  the  arc,  and  if  produced,  falls  entirely  without  the 
curve. 

This  property  was  separately  proved  for  the  circle  by 
Euclid,  for  the  conic  sections  by  Apollonius,  for  cylinders 
by  Serenus. 

4.  Curves   of  this   sort   are   convex  or   concave   to  the 
same  parts  throughout. 

This  is  the  same  as  saying  that  a  straight  line  only  cuts 
the  curve  in  two  points ;  nor  does  it  differ  from  the  definition 
of  "  hollow,"  as  given  by  Archimedes  at  the  beginning  of 
his  book  on  the  sphere  and  the  cylinder. 

5.  All  straight  lines  parallel  to  the  genetrix  cut  the  curve ; 
and  any  one  cuts  the  curve  in  one  point  only. 

This  was  proved,  separately,  for  the  parabola  and  the 
hyperbola  by  Apollonius,  and  for  the  sections  of  the 
concoids  by  Archimedes. 

6.  Similarly  all  parallels  to   the  directrix  cut  the  curve, 
and  in  one  point  only. 

Apollonius  proved  this  for  the  sections  of  the  cone. 

7.  All  chords   of  the   curve  meet   the   genetrix  and  all 
parallels  to  it,  produced  if  necessary. 

Apollonius  thought  it  worth  while  to  prove  this  property 
separately  for  the  parabola  and  the  hyperbola. 

8.  Similarly,  all  straight  lines  touching   the   curve,  with 
one  exception  (see  §  1 8),  meet  the  same  parallels. 

This  also  Apollonius  showed  for  the  conic  sections  in 
separate  theorems. 


LECTURE  IV  55 

9.  Also  any  straight  lines  cutting  the  genetrix  will  also 
cut  the  curve. 

Apollonius  went  to  a  very  great  deal  of  trouble  to  prove 
a  property  of  this  kind  in  the  case  of  the  conic  sections. 

10.  Straight  lines  applied  to  the   directrix,  i.e.  parallels 
to  the  genetrix,  have  a  ratio  to  one  another  (when  the  less 
is  the  antecedent)  which  is  less  than  the  ratio  of  the  corre- 
sponding spaces  traversed  by  the  moving  straight  line ;  i.e. 
the  ratio  of  the  versed  sines  of  the  curve,  the  less  to  the 
greater,  is  less  than  the  ratio  of  the  sines. 

This  property  of  circles  and  other  curves,  it  will  be  found, 
is  everywhere  proved  separately  for  each  kind. 

NOTE 

All  the  preceding  properties  are  deduced  in  a  very  simple 
manner  from  one  diagram ;  and  Barrow's  continual  claim 
that  his  method  not  only  simplifies  but  generalizes  the 
work  of  the  early  geometers  is  substantiated. 

The  full  proof  of  the  next  property  is  given,  to  illustrate 
Barrow's  way  of  using  one  of  the  variants  of  the  method  of 
exhaustions. 

11.  Let  us  suppose  that  a  straight  line  IMS  touches  a 
given  curve  at  a  point  M  (i.e.  it 

does   not   cut   the    curve) ;   and 

let  the   tangent  meet   AZ   in   T,    A — 

and  through  M  let  PMG  be  drawn 

parallel  to  AY.     I  may  say  that    p 

the   velocity   of  the   descending 

point,  describing  the  curve  by  its    Z 

motion,  which  it  has  at  the  point  Fig.  20. 


o 


K 
\S 


° 


K 

\ 

\S 


56  BARROWS  GEOMETRICAL  LECTURES 

of  contact  M,  is  equal  to  the  velocity  by  which  the 
straight  line  TP  would  be  described  uniformly,  in  the 
same  time  as  the  straight  line 
AZ  is  carried  along  AC  or  PM  (or, 
what  comes  to  the  same  thing,  I  A 

say  that  the  velocity  of  the  de- 

p 
scending  point  at  M  has  the  same 

ratio  to  the  velocity  with  which 
the  straight  line  AZ  is  moving  as 
the  straight  line  TP  has  to  the  straight  line  PM). 

For,  let  us  take  anywhere  in  the  tangent  TM  any  point 
K,  and  through  it  draw  the  straight  line  KG,  meeting  the 
curve  in  0  and  the  parallels  AY,  PG  in  D,  G.     Then,  since 
the  tangent  TM  is  supposed  to  be  described  by  a  twofold 
uniform    motion,    partly   of  the    straight    line   TZ   carried 
parallelwise  along  AC   or  PM,  and   partly  of  the  point  T 
descending  from  T  along  TZ ;  and  since,  of  these  motions, 
the  one  along  AC  or  PM  is  common  with,  or  the  same  as, 
that  by  which  the  curve  is  described ;  and  since,  when  TZ 
is  in  the  position  KG,  AZ  will  be  in  the  same  position  as 
well ;  therefore,  when  the  point  descending  from  T  is  at  K, 
the  point  descending  from  A  will  be  at  0,  the  intersection 
of  the  curve  with  KG  (for  the  straight  line  KG  cannot  cut 
the  curve  in  any  other  point,  as  has  already  been  shown). 
Also  the  point  0  is  below  K,  because  the  tangent  lies  en- 
tirely outside  the  curve.     Now,  if  the  point  0  is  supposed 
to  be  above  the  point  of  contact,  towards  T,  since  in  that 
case  OG  is  less  than  GK,  it  is  clear  that  the  velocity  of  the 
descending  point,  by  which  the  curve  is  described,  at  the 


LECTURE  IV  57 

point  0  is  less  than  the  velocity  of  the  uniformly  descend- 
ing motion,  by  which  the  tangent  is  produced ;  since  the 
former,  always  increasing,  in  the  same  time  (namely  that 
represented  by  GM)  traverses  a  smaller  space  than  the  latter 
which  does  not  increase  at  all ;  and  as  this  goes  on  continu- 
ally, the  former  describes  the  straight  line  OG  whilst  the 
latter  describes  the  straight  line  KG.  On  the  other  hand, 
if  the  point  K  is  below  the  point  of  contact  towards  the  end 
8,  since  OG  is  then  greater  than  KG,  it  is  clear  that  the 
velocity  of  the  descending  point,  by  which  the  curve  is  pro- 
duced, at  the  point  0,  in  the  same  way  as  before,  is  greater 
than  the  velocity  of  the  uniformly  descending  motion,  by 
which  the  tangent  is  described;  for  the  former  motion, 
continually  decreasing  during  the  time  represented  by  GM, 
traverses  a  greater  space  than  the  latter,  which  does  not 
decrease  at  all,  but  keeping  constant,  describes  indeed  the 
space  KG.  Hence,  since  the  velocity  of  the  point  describ- 
ing the  curve,  at  any  point  of  the  curve  above  the  point  of 
contact  towards  A,  is  less  than  the  velocity  of  the  motion 
for  TP ;  and  at  any  point  of  the  curve  below  the  point  of 
contact  is  greater  than  it ;  it  follows  that  it  is  exactly  equal 
to  it  at  the  point  M. 

12.  The  converse  of  the  preceding  theorem  is  also  true. 

13.  From  these   two   theorems,  it  follows   at   once  that 
curves  of  this  kind  are  touched  by  any  one  straight  line  in 
one  point  only. 

This,  separately,  Euclid  proved  for  the  circle,  Apollonius 
for  the  conic  sections,  and  others  for  other  curves. 


58  HARROW'S  GEOMETRICAL  LECTURES 

From  this  method,  then,  there  comes  out  an  advantage 
not  to  be  despised,  that  by  the  one  piece  of  work  proposi- 
tions are  proved  concerning  tangents  in  several  cases. 

14.  The   velocities  of  the  descending  point  at  any  two 
assigned  points  of  a  curve  have  to  one  another  the  ratio 
reciprocally  compounded  from  the  ratios  of  the  lines  applied 
to  the  straight  line  AZ  from  these  points  (i.e.  parallels  to 
AY)   and  the  intercepts    by  the   tangents   at   these   points 
measured  from  the  said  applied  lines.     In  other  words,  the 
ratio  of  the  velocities  is  equal  to  the  ratio  of  the  intercepts 
divided  by  the  ratio  of  the  applied  lines. 

15.  Incidentally,  I  here  give  a  general  solution  by  my 
method,  and  one   quite   easy  to   follow,  of  that   problem 
which  Galileo  made  much  of,  and  on  which  he  spent  much 
trouble,  about  which  Torricelli  said  that  he  found  it  most 
skilful  and  ingenious.     Torricelli  thus  enunciates  it  (for  the 
enunciation  of  Galileo  is  not  at  hand) : — 

"  Given  any  parabola  with  vertex  A,  it  is  required  to  find 
some  point  above  it,  from  which  if  a  heavy  body  falls  to  A, 
and  from  A,  with  the  velocity  thus  attained,  is  turned  hori- 
zontally, then  the  body  will  describe  the  parabola." 

NOTE 

Barrow  gives  a  very  easy  construction  for  the  point,  and 
a  short  simple  proof;  further  his  construction  is  perfectly 
general  for  any  curve  of  the  form  y  =  xn,  where  n  is  a  posi- 
tive integer. 

16.  17.  These  are  two  ingenious  methods  for  determining 
the  ratio  of  the  abscissa  to  the  subtangent. 

Barrow  remarks  that  the  theorems  will  be  proved  more 
geometrically  later,  so  that  they  need  not  be  given  here. 


LECTURE  IV  59 

1 8.  A  circle,  an  ellipse,  or  any  "returning"  curve  of  this 
kind,  being  supposed  to  be  generated  by  this  method,  then 
the  point  describing  any  one  of  them  must  have  an  infinite 
velocity  at  the  point  of  return. 

For  instance,  let  a  quadrant  APM  be  so  generated;  then 
since  the  tangent,  TM,  is  parallel  to  the  diameter  AZ,  and 
only  meets  it  at  an  infinite  distance,  therefore  the  velocity 
at  M  is  to  the  velocity  of  the  uniform  motion  of  AZ  parallel 
to  itself  as  an  infinite  straight  line  is  to  PM ;  hence,  the 
velocity  at  M  must  certainly  be  infinite.  And  indeed  it 
will  be  so  for  all  curves  of  this  kind ;  but  for  others  which 
are  gradually  continued  to  infinity  (such  as  the  parabola 
or  the  hyperbola)  the  velocity  of  the  descending  point  at 
any  point  on  the  curve  is  finite. 

Leaving  this,  let  us  go  on  to  those  other  properties  of  the 
given  curves  which  have  to  be  expounded. 

NOTE 

It  is  to  be  observed  that,  although  Barrow  usually  draws 
his  figures  with  the  applied  lines  at  right  angles  to  his 
directrix  AZ,  his  proofs  equally  serve  if  the  applied  lines  are 
oblique,  in  all  cases  when  not  otherwise  stated.  That  is, 
analytically,  his  axes  may  be  oblique  or  rectangular.  Having 
mentioned  this  point,  since  my  purpose  is  largely  with 
Barrow's  work  on  the  gradient  of  the  tangent,  I  shall  always 
draw  the  applied  lines  at  right  angles,  as  Barrow  does ;  ex- 
cept in  the  few  isolated  cases  where  Barrow  has  intentionally 
drawn  them  oblique. 


LECTURE   V 


Further  properties  of  curves.  Tangents.  Curves  like  the 
Cycloid.  Normals.  Maximum  and  minimum  lines. 

i .  The  angles  made  with  the  applied  lines  by  the  tangents 
at  different  points  of  a  curve  are  unequal ;  and  those  are 
less  which  are  nearer  to  the  point  A,  the  vertex. 


Fig.  26. 


2.  Hence   it  may  be  taken   as   a   general   theorem  that 
tangents  cut  one  another  between  the  applied  lines  drawn 
at  right  angles  to  AZ  through  the  points  of  contact. 

3.  The  angle  PTM  is  greater  than  the  angle  XQN. 

4.  Applied  lines  nearer  to  the  vertex  (and  therefore  also 
any  straight  lines  parallel  to  other  directions)  cut  the  curve 
at  a  greater  angle  than  those  more  remote. 


LECTURE    V  61 

5.  If  the  angle  made  by  an  applied  line  is  a  right  angle 
or  obtuse,  I  say  that  the  arc  MN  of  the  curve  is  greater 
than  the  straight  line  MN,  but  less  than  the  straight 
line  ME. 

This  is  a  most  useful  theorem  for  service  in  proving 
properties  of  tangents.  For,  it  follows  from  it  that,  if  the 
arc  MN  is  assumed  to  be  indefinitely  small^  we  may  safely 
substitute  instead  of  it  the  small  bit  of  the  tangent ',  i.e.  either 
MEarNH, 

NOTE 

We  have  here  the  statement  of  the  fundamental  idea  of 
Barrow's  method,  to  which  all  the  preceding  matter  has  led. 
This  is  a  fine  illustration  of  Barrow's  careful  treatment ;  and 
it  is  to  be  observed  that  this  idea  is  not  quite  the  same  thing 
as  the  idea  of  the  differential  triangle  as  one  is  accustomed 
to  consider  it  nowadays,  i.e.  as  a  triangle  of  which  the  hypo- 
tenuse is  an  infinitely  small  arc  of  the  curve  that  may  be 
considered  to  be  a  straight  line.  It  will  be  found  later  that 
Barrow  uses  the  idea  here  given  in  preference  to  the  other, 
because  by  this  means  the  similarity  of  the  infinitesimal 
triangle  with  the  triangle  TPM  is  far  more  clearly  shown  on 
his  diagrams ;  and  many  matters  in  Barrow  are  made  sub- 
servient to  this  endeavour  to  attain  clearness  in  his  diagrams. 
For  instance,  when  he  divides  a  line  into  an  infinite  number 
of  parts,  he  generally  uses  four  parts  on  his  figure,  and  gives 
the  demonstration  with  the  warning  "on  account  of  the 
infinite  division  "  as  a  preliminary  statement. 

As  an  example  of  the  use  to  which  the  above  theorems 
may  be  put,  Barrow  finds  the  tangent  to  the  Cycloid,  his 
construction  being  applicable  to  all  curves  drawn  by  the 
same  method.  Note  that  this  is  not  the  general  case  of  the 
roulette  discussed  by  Descartes.  Barrow's  construction  and 
proof  are  given  in  full  to  bring  out  the  similarity  of  his 
criterion  of  tangency  to  Fermat's  idea,  as  mentioned  in 
the  Introduction. 


62  BARROW'S  GEOMETRICAL  LECTURES 

6.  A  straight  line  AY,  moving  parallel  to  itself,  traverses 
any  curve,  either  concave  or  convex  to  the  same  parts, 
with  uniform  motion  (that  is  to  say,  it  passes  over  equal 
parts  of  the  curves  in  equal  times),  and  simultaneously 
any  point  is  carried,  also  uniformly,  along  AY  from  A ; 
by  the  point  moving  in  this  manner  there  is  generated 
a  curve  AMZ,  of  which  it  is  required  to  find  the  tangent 
at  any  point  M. 


\H 


Fig.  27. 

To  do  this,  draw  MP  parallel  to  AY  to  cut  the  curve  APX 
in  P;  through  P  draw  the  straight  line  PE  touching  the 
curve  APX;  through  M  draw  MH  parallel  to  PE;  take  any 
point  R  in  MH,  and  draw  RS  parallel  to  PM  ;  mark  off  RS 
so  that  MR  :  RS  =  arc  AP  :  PM  (i.e.  as  the  one  uniform 
motion  is  to  the  other);  join  MS.  Then  MS  will  touch 
the  curve  AMZ. 

For,  if  any  point  Z  be  taken  in  this  curve,  and  through 
it  ZK  be  drawn  parallel  to  MP,  cutting  the  curve  APX  in  X, 
the  tangent  at  P  in  E,  MH  the  parallel  to  it  in  H,  and  MS 
in  S ;  then, 

(i),  if  the  point  Z  is  above  M  towards  A,       PE  <  arc  PX  ; 
.'.  arc  PA  :  PE  >  arc  PA  :  arc  PX. 


LECTURE    V  63 

But  arc  PA  :  arc  PX  =  PM  :  PM  -  XZ  =  PM  :  EH  -  XZ 

arc  PA  :  arc  PX  =  PM  :  ZH  -  EX  >  PM  :  ZH  ; 
hence,  arc  PA  :  PE  >  PM  :  ZH     or    arc  PA  :  PM  >  PE  :  ZH. 

But  arc  PA  :  PM  =  MR  :  RS  =  MH  :  KH  =  PE  :  KH  ; 
PE:KH  >  PE:ZH,     and     KH  <  ZH. 

Now,  since  EZ  <  XZ  <  PM  or  EH,  the  point  H  is  outside 
the  curve  AZM  ;  hence  K  is  outside  the  curve  AZM. 

Similarly  [Barrow  gives  it  in  full],  (ii),  if  the  point  Z  is 
below  the  point  M,  K  will  be  outside  the  curve;  therefore 
the  whole  straight  line  KMK8  lies  outside  the  curve,  and 
thus  touches  it  at  M. 

After  this  digression  we  will  return  to  other  properties  of 
the  curve. 

7.  Any  parallel  to  the  tangent  TM,  through  a  point  E 
directly  below  T,  will  meet  the  curve.     [Fig.  26.] 

8.  If  E  lies  between  the  point  T  and  the  vertex  A,  the 
parallel  to  the  tangent  will  cut  the  curve  twice. 

Apollonius  was  hard  put  to  it  to  prove  these  two  theorems 
for  the  conic  sections. 

9.  If  any  two  lines  are   equally  inclined  to  the  curve, 
these  straight  lines  diverge  outwardly,  i<c.  they  will  meet 
one  another  when  produced   towards  the  parts  to  which 
the  curve  is  concave. 

10.  If  a  straight  line  is  perpendicular  to  a  curve,  and 
along  it  a  definite  length   HM   is  taken,  then   HM   is  the 
shortest  of  all  straight  lines  that  can  be  drawn  to  the  curve 
from  the  point  H. 


64  BARROWS  GEOMETRICAL  LECTURES 

11.  It   follows   that   the    circle,    with    centre    H,    drawn 
through  M,  touches  the  curve. 

12.  Conversely,  if  HM  is  the  shortest  of  all  straight  lines 
that  can  be  drawn  from  H  to  the  curve,  then  H  M  will  be 
perpendicular  to  the  curve. 

13.  If  HM  is  the  shortest  of  all  straight  lines  that  can  be 
drawn  from  H,  and  if  the  straight  line  TM  is  perpendicular 
to  it,  then  TM  touches  the  curve. 

14.  Further,  a  line  which  is  nearer  to  HM  is  shorter  than 
one  which  is  more  remote. 

15.  Hence  it  follows  that  any  circle  described  with  centre 
H  meets  the  curve  in  one  point  only  on  either  side  of  M  ; 
that  is,  it  does  not  cut  the  curve  in  more  than  two  points 
altogether. 

1 6.  If  two  straight  lines  are  parallel  to  a  perpendicular, 
the  nearer  of  these  will  fall  more  nearly  at  right  angles  to 
the  curve  than  the  one  more  remote. 

17.  If  from   any  point   in   the   perpendicular    HM,   two 
straight  lines  are  drawn  to  the  curve,  the  nearer  will  fall 
more  nearly  at  right  angles  to  the  curve  than  the  one  more 
remote. 

1 8.  Hence   it   is   evident    that    by   moving   away   from 
the    perpendicular,    the    obliquity    of    the    incident    lines 
with   the   curve   increases,  until    that   which   touches   the 
curve  is  reached ;   this,  the  tangent,  is  the  most  oblique 
of  all. 

19.  If  the  point  H  is  taken  within  the  curve,  and  if,  of 


LECTURE    V  65 

all  lines  drawn  from  it  to  meet  the  curve,  HM  is  the  least ; 
then  HM  will  be  perpendicular  to  the  curve  or  the  tangent 
MT. 

20.  Also,  if  HM  is  the  greatest  of  all  straight  lines  drawn 
to  meet  the  curve,  then  HM  will  be  perpendicular  to  the 
curve. 

21.  Hence,  if  MT  is  perpendicular  to  HM,  whether  the 
latter  is  a  maximum  or  a  minimum,  it  will  touch  the  curve. 

22.  It  follows  that,  if  a  straight  line  is  not  perpendicular 
to  the  curve,  no  greatest  or  least  can  be  taken  in  it. 

23.  If  HM  is  the  least  of  the  lines  drawn  to  the  curve,  and 
any  point  I  is  taken  in  it;  then  IM  will  be  a  minimum. 

24.  If  HM  is  the  greatest  of  the  lines  drawn  to  meet  the 
curve,  and  any  point  I  is  taken  on  MH  produced;  then  IM 
will  be  a  maximum. 

For  the  rest,  the  more  detailed  determination  of  the 
greatest  and  least  lines  to  a  curve  depends  on  the  special 
nature  of  the  curve  in  question 

[Barrow  concludes  these  preliminary  five  lectures  with 
the  remark  : — ] 

"  But  I  must  say  that  it  seems  to  me  to  be  wrong,  and 
not  in  complete  accord  with  the  rules  of  logic,  to  ascribe 
things  which  are  applicable  to  a  whole  class,  and  which 
come  from  a  common  origin,  to  certain  particular  cases,  or 
to  derive  them  from  a  more  limited  source." 

NOTE 
The  next  lecture  is  the  first  of  the  seven,  as  originally 

5 


66    BARROW'S  GEOMETRICAL  LECTURES 

designed,  that  were  to  form  a  supplement  to  the  Optics. 
Barrow  begins  thus  : — 

"  I  have  previously  proved  a  number  of  general  properties 
of  curves  of  continuous  curvature,  deducing  them  from  a 
certain  mode  of  construction  common  to  all ;  and  especially 
those  properties,  as  I  mentioned,  that  had  been  proved  by 
the  Ancient  Geometers  for  the  special  curves  which  they 
investigated.  Now  it  seems  that  I  shall  not  be  displeasing, 
if  I  shall  add  to  them  several  others  (more  abstruse  indeed, 
but  not  altogether  uninteresting  or  useless) ;  these  will  be, 
as  usual,  demonstrated  as  concisely  as  possible,  yet  by  the 
same  reasoning  as  before ;  this  method  seems  to  be  in  the 
highest  degree  scientific,  for  it  not  only  brings  out  the  truth 
of  the  conclusions,  but  opens  the  springs  from  which  they 
arise. 

The  matters  we  are  going  to  consider  are  chiefly  con- 
cerned with 

(i)  An  investigation  of  tangents,  freed  from  the  loathsome 
burden  of  calculation,  adapted  alike  for  investigation  and 
proof  (by  deducing  the  more  complex  and  less  easily  seen 
from  the  more  simple  and  well  known) ; 

(ii)  The  ready  determination  of  the  dimensions  of  many 
magnitudes  by  the  help  of  tangents  which  have  been 
drawn. 

These  matters  seem  not  only  to  be  somewhat  difficult 
compared  with  other  parts  of  Geometry,  but  also  they  have 
not  been  as  yet  wholly  taken  up  and  exhaustively  treated 
(as  the  other  parts  have) ;  at  the  least  they  have  not  as  yet 
been  considered  according  to  this  method  that  2  know.  So  we 


LECTURE    VI  67 

will  straightway  tackle  the  subject,  proving  as  a  preliminary 
certain  lemmas,  which  we  shall  see  will  be  of  considerable 
use  in  demonstrating  more  clearly  and  briefly  what  follows." 


The  original  opening  paragraph  to  the  "  seven  "  lectures 
probably  started  with  the  words,  "  The  matters  we  are  going 
to  consider,  etc.,"  the  first  paragraph  being  afterwards 
added  to  connect  up  the  first  five  lectures. 

Barrow  indicates  that  his  subject  is  going  to  be  the  con- 
sideration of  tangents  in  distinction  to  the  other  parts  of 
geometry,  which  had  been  already  fairly  thoroughly  treated  ; 
he  probably  alludes  to  the  work  on  areas  and  volumes  by 
the  method  of  exhaustions  and  the  method  of  indivisibles, 
of  which  some  account  has  been  given  in  the  Introduction ; 
when  he  treats  of  areas  and  volumes  himself,  he  intends 
to  use  the  work  which,  by  that  time,  he  has  done  on 
the  properties  of  tangents.  From  this  we  see  the  reason 
why  the  necessity  arose  for  his  two  theorems  on  the  inverse 
nature  of  differentiation  and  integration. 

That  Barrow  himself  knew  the  importance  of  what  he 
was  about  to  do  is  perfectly  evident  from  the  next  para- 
graph. He  distinctly  says  that  Tangents  had  been  investi- 
gated neither  thoroughly  nor  in  general;  also  he  claims 
distinctly  that,  to  the  best  of  his  knowledge  and  belief,  his 
method  is  quite  original.  He  further  suggests  that  it  will  be 
found  a  distinct  improvement  on  anything  that  had  been 
done  before.  In  other  words,  he  himself  claims  that  he  is 
inventing  a  new  thing,  and  prepares  to  write  a  short  text- 
book on  the  Infinitesimal  Calculus.  And  he  succeeds,  no 
matter  whether  the  style  is  not  one  that  commended  itself 
to  his  contemporaries,  or  whether  the  work  of  Descartes 
had  revolutionized  mathematical  thought;  he  succeeds  in 
his  task.  In  exactly  the  same  way  as  the  man  who  put  the 
eye  of  a  needle  in  its  point  invented  the  sewing-machine. 

Barrow  sets  out  with  being  able  to  draw  a  tangent  to  a 
circle  and  to  a  hyperbola  whose  asymptotes  are  either  given 
or  can  be  easily  found,  and  the  fact  that  a  straight  line  is 
everywhere  its  own  tangent.  Whenever  a  construction  is 
not  immediately  forthcoming  from  the  method  of  description 


68    BARROW'S  GEOMETRICAL  LECTURES 

of  the  curve  in  hand,  he  usually  has  some  means  of  drawing 
a  hyperbola  to  touch  the  curve  at  any  given  point ;  he  finds 
the  asymptotes  of  the  hyperbola,  and  thus  draws  the  tangent 
to  it ;  this  is  also  a  tangent  to  the  curve  required.  Analyti- 
cally, for  any  curve  whose  equation  is  v  =  f(x),  he  uses  as  a 
first  approximation  the  hyperbola  y  =  (ax  +  fr)l(cx  +  d). 

He  then  gives  a  construction  for  the  tangent  to  the 
general  paraboliform,  and  makes  use  of  these  curves  as 
auxiliary  curves.  As  will  be  found  later,  he  proves  that 
i  +  nx  is  an  approximation  to  ( i  +  x)n,  leading  to  the 
theorem  that  if  y  =  xn,  then  dyjdx  =  n  .y/x.  Thus  he  founds 
the  whole  of  his  work  on  exactly  the  same  principles  as 
those  on  which  the  calculus  always  is  founded,  namely,  on 
the  approximation  to  the  binomial  theorem ;  and  he  does 
it  in  a  way  that  does  not  call  for  any  discussion  of  the  con- 
vergence of  the  binomial  or  any  other  series. 

For  the  benefit  of  those  who  are  beginners  in  mathe- 
matical history,  it  may  not  be  out  of  place  if  I  here  reiterate 
the  warning  of  the  Preface  (for  Prefaces  are  so  often  left 
unread)  that  Barrow  knew  nothing  of  the  Calculus  notation  of 
Leibniz.  Barrow's  work  is  geometrical,  as  far  as  his  published 
lectures  go ;  the  nearest  approach  to  the  calculus  of  to-day 
is  given  in  the  "<2  and  e  "  method  at  the  end  of  Lecture  X. 

Again,  with  regard  to  the  differentiation  of  the  com- 
plicated function,  given  as  a  specimen  at  the  end  of  this 
volume,  I  do  not  say  that  Barrow  ever  tackled  such  a  thing. 
What  I  do  urge,  however,  is  that  Barrow  could  have  done 
so,  if  he  had  come  across  such  a  function  in  his  own  work. 
My  argument,  absolutely  conclusive  I  think,  is  that  I  have 
been  able  to  do  so,  using  nothing  but  Barrow's  theorems 
and  methods. 


LECTURE    VI 

Lemmas ;  determination  of  curves  constructed  according  to 
given  conditions ;  mostly  hyperbolas. 

1.  [The  opening  paragraph,  as  quoted  in  the  note  at  the 
end  of  the  preceding  lecture.] 

2.  Let  ABC  be  a  given  angle  and  D  a  given  point;  also 
let  the  line  ODO  be  such  that,   if  any  straight  line  DN  is 
drawn  through  D,  the  length  MM,  intercepted  between  the 
arms  of  the  angle,  is  equal  to  the  length  DO,  intercepted 
between  the  point  D  and  the  line  ODO;  then  the  line  ODO 
will  be  a  hyperbola.* 

Moreover,  if  MN  is  supposed  to  bear  always  the  same 
ratio  to  DO  (say  a  given  ratio  R  :  8),  the  line  ODO  will  be  a 
hyperbola  in  this  case  also. 

3.  Here   I  note,   in  passing,  that  it  is  easy  to  solve  the 
problem  by  which  the  solutions  of  the  problems  of  Archi- 
medes and  of  Vieta   were   reduced  to  conic   sections  by 
the  aid  of  a  previously  cc-nstructed  conchoid. 

Thus  "  to  draw  through  a  given  point  D  a  straight  line,  so 

*  There  is  a  very  short  proof  given  to  this  theorem,  as  an  alternative. 
It  is  hard  to  see  why  the  comparatively  clumsy  first  proof  is  retained,  unless 
the  alternative  proof  was  added  in  revise  (?  by  Newton).  There  is  also 
a  reference  to  easy  alternative  proofs  for  §§  4,  9.  These  alternative  proofs 
depend  on  an  entirely  different^  property  of  the  curve. 


70   BARROW'S   GEOMETRICAL  LECTURES 

that  the  part  of  the  straight  line  so  drawn,  intercepted 
between  the  arms  of  a  given  angle  ABC,  may  be  equal  to  a 
given  straight  line  T." 

For,  if  the  hyperbola  (of  the  preceding  article)  is  first 
described,  and  if  with  centre  D,  and  a  radius  equal  to  the 
given  straight  line  T,  a  circle  POQ  is  described,  cutting  the 
hyperbola  in  0,  and  DO  is  produced  to  cut  the  arms  of 
the  angle  in  M  and  N  ;  then  it  follows  that  MN  =  DO  =  T. 

4.  Let  ABC  be  a  given  angle  and  D  a  given  point;  and 
let  the  line  OBO  be  such  that,  if  through  D  any  straight 
line  DN  is  drawn,,  the  length  MN  intercepted  between  the 
arms  of  the  angle  bears  always  the  same  ratio  (say  X  :  Y) 
to  the  length  MO  intercepted  between  the  arm  BC  and  the 
curve  OBO;  then  OBO  will  be  a  hyperbola. 

5-  If  MO  is  taken  on  the  other  side  of  the  straight  line 
BC,  the  method  of  proof  is  the  same. 

6.  INFERENCE. — If  a  straight  line   BQ  divides  the  angle 
ABC,  and  through  the  point  D  are  drawn,  in  any  manner, 
two  straight  lines  MN,  XY,  cutting  the  straight  line  BQ  in 
the  points  0,  P,  of  which  0  is  the  nearer  to  B;  then 

MN:MO  <  XY:XP. 

7.  Moreover,    if    several    straight    lines     BQ,     BG   .   .  . 
divide  the  angle  ABC,  and  if  from  the  point  D  the  straight 
lines  DN,  DY  are  drawn,  cutting  BC,  BQ,  BG,  BA  in  M,  0, 
E,    N    and   X,  V,  F,  Y,  DN   being   the   nearer   to  B ;  then 
NE:MO<  YF:VX. 

8.  From  what  has  gone  before,  it  is  also   evident  that 
through  B  (in  one  of  two  directions)  a  straight  line  can  be 


LECTURE   VI  71 

so  drawn  that  the  segments  intercepted  on  lines  drawn 
through  D  between  the  constructed  line  and  BC  shall 
have  to  the  segments  intercepted  between  BA  and  BC  a 
ratio  that  is  less  than  a  given  ratio. 

9.  Again,  suppose  a  given  angle  ABC  and  a  given  point 
D ;  also  let  the  line  000  be  such  that,  if  through  D  any 
straight  line  DO  is  drawn,  cutting  the  arms  of  the  angle 
in  M,  N,  then  DM  always  bears  to  NO  a  given  ratio  (X  :  Y  say;) 
then  the  line  000  will  be  a  hyperbola. 

10.  A  straight  line   ID  being  given  in  position,  and  a 
point   D  fixed  in  it,  let  DNN  be  a  curve  such  that,  if  any 
point  G   is  taken   in    ID,   and  a  straight  line  GN  is  drawn 
parallel  to  a  straight  line  IK  given  in  position,  and  if  two 
straight  lines  whose  lengths  are  m  and  b  are  taken,  and  if 
we  put  DG  =  x,  and  GN  =  y,  there  is  the  constant  relation 
my  +  xy  =  mx^jb',  then  DNN  will  be  a  hyperbola. 

11.  If  the  equation  is  my  —  xy  —  mx*jb,  the  same  hyper- 
bola is  obtained,  only  G  must  be  taken  in  DM  instead  of 
DO.     If,  however,  the  equation  is   xy  —  my  =  mx2/fr,  then 
G  must  be  beyond  M  and  the  hyperbola  conjugate  to  the 
former  is  obtained. 

12.  If  BDF  is  a  given  triangle  and  the  line  DNN  is  such 
that,  if  any  straight  line  RN  is  drawn  parallel  to  BD,  cutting 
the  lines  BF,   DF,   DNN  in  the  points  R,  G,  N,  and  DN  is 
joined;    and   if  DN   is   then   always    a   mean    proportional 
between  RN  and  NG  ;  then  the  line  DNN  is  a  hyperbola. 

13.  If  ID  is  a  straight  line  given  in  position;  and  DNN  is 
a  curve   such   that,  if  any  point  G  is  taken  in  ID,  and  the 


72    BARROWS  GEOMETRICAL  LECTURES 

straight  line  GN  is  drawn  parallel  to  IK,  a  straight  line  given 
in  position,  and  if  straight  lines  whose  lengths  are  g,  m,  r, 
are  taken ;  and  if  we  put  DG  =  x,  and  GN  =  y,  then  there  is 
a  constant  relation  xy+gx-mv  =  mx^/r;  then  the  line 
DNN  will  be  a  hyperbola. 

If  the  equation  is  -yx+gx  +  my  ~  mxz/b,  then  the  same 
hyperbola  is  obtained,  but  the  points  G  must  then  be  taken 
between  B  and  M  (B  being  the  point  where  the  curve  cuts 
the  straight  line  ID);  and  if  the  points  G  are  assigned  to 
other  positions,  the  signs  of  the  equation  vary.  But  it  is 
not  opportune  to  go  into  them  at  present. 

14.  Two  straight  lines  DB,  DA,  are  given  in  position,  and 
along  the  line  DB  a  straight  line  CX  is  carried  parallel  to 
BA  ;  also,  by  turning  round  the  point  D  as  a  centre,  a  straight 
line  DY  moves  so  that,  if  it  cuts  BA  in  X,  there  is  always 
the  same  ratio  between  the  lines  BE  and  CD  (equal  to  the 
ratio  of  some  assigned  length   R  to  DB,  say);  then,  if  DE 
cuts  CX  in  N,  the  line  DNN  is  a  parabola. 

Gregory  St  Vincent  gave  this,  but  demonstrated  with 
laborious  prolixity,  if  I  remember  rightly. 

We  add  the  following:— 

15.  If,  other  things  remaining  the  same,  CX  and  DY  are 
moved  in  such   a  way  that  now   BE  and    BC  are   always 
in  the   same    ratio  (BD :  R,    say);    their    intersections    will 
give  a  parabola  also. 

1 6.  If,  with  other  things  remaining  the  same,  the  straight 
line  CX  is    not  now  carried  parallel  to   BA,   but  to  some 
other  straight  line  DH,  given  in  position ;  and  if  the  ratio 


LECTURE   VI  73 

of  BE  to  DC  is  always  equal  to  the  ratio  of  DB  to  R  ;  then 
the  intersections  N  will  lie  on  a  hyperbola. 

17.  Moreover,   other  things  remaining  the  same  as  in 
the  preceding,  if  CX  now  moves  in  such  a  way  that  BE 
always  bears  the  same  ratio  to  BC  (BD  :  R,  say) ;  the  inter- 
sections in  this  case  will  also  lie  on  a  hyperbola. 

1 8.  Let  two  straight  lines  DB,  DA  be  given  in  position, 
and  a  point  D  fixed  in  DB;  and  let  the  line  DNN  be  such 
that,  if  any  straight  line  GN  is  drawn  parallel  to  BA,  and 
two  straight  lines  whose  lengths  are  ^,  r  are  taken,  and  DG, 
GN  are  called  x,  y;  and  if  ry  —  xy  —  gx\  then  the  line  DNN 
will  be  a  hyperbola. 

If,  however,  the  equation  is  xy  -  ry  =  gx,  we  must  take 
DE  =  r,  and  BO  =  g  (measured  below  the  line  DB);  the 
proof  is  the  same  as  before. 

19.  Let  two  straight  lines  DB,  BA  be  given  in  position; 
and  let  the  straight  line  FX  move  parallel  to  DB,  and  let  DY 
pass  through  the  fixed  point  D,  so  that  the  ratio  of  BE  to 
BF  is  always  equal  to  an  assigned  ratio,  say  DB  to  R;  then 
the   intersections   of  the   straight   lines    DY,  GN    lie   on  a 
straight  line. 

20.  But  if,  other  things  remaining  the  same,  some  other 
point  0  is  taken  in  AB,  which  we  take  as  the  origin  of  reckon- 
ing, so  that  the  ratio  BE  to  OF  is  always  equal  to  the  ratio 
DB  to  R ;  then  the  intersections  will  lie  on  a  hyperbola. 

21.  Moreover,  other  things  remaining  the  same,  let  the 
straight  line  FX  now  move  not  parallel  to  DB,  but  to  another 
straight   line  DH,  so  that,  a  fixed  point  0  being  taken  in 


74    BARROW'S  GEOMETRICAL  LECTURES 

BA,  the  ratio  BE  to  OF  is  always  equal  to  an  assigned  ratio 
(say  DB  to  m);  then  the  intersections  will  again  lie  on  a 
hyperbola. 

22.  Let  ADB  be  a  triangle  and  DYY  a  line  such  that,  if 
any  straight  line  PM  is  drawn  parallel  to  DB,  meeting  AB 
in  M,  PY  is  always  equal  to   ^(PM2  -  DB2);  then  the  line 
DYY  is  a  hyperbola. 

COR.  If  YS  is  the  tangent  to  the  hyperbola  DYY,  then 
PM2 :  PY2  =  PA  :  PS. 

23.  If,  other   things  remaining  the  same,  we  have  now 
PY  =  V(PM2  +  DB2);  then  the  line  DYY  is  again  a  hyperbola. 

COR.  If  YS  is  the  tangent  to  the  hyperbola,  then 
PM2  :  PY2  =  PA  :  PS. 

24.  If  ADB  is  a  triangle,  having  the  angle  ADB  a  right 
angle,  and  the  curve  CGD  is  such  that,  if  any  straight  line 
PEG  is  drawn  parallel  to  DB,  cutting  the  sides  of  the  triangle 
in  F,  E,  and  the  curve  in  G,  the  rectangle  contained  by  EF 
and  EG  is  equal  to  the  square  on  DB;  then  the  curve  CGD 
is  an  ellipse,  of  which  the  semi-axes  are  AD,  AC. 

COR.  Let  GT  be  a  tangent  to  the  ellipse,  then 
EF2:EG2  =  AE:AT. 

25.  If  DTH  is  any  rectilineal  angle,  and  A  is  a  fixed  point 
in  TD,  one  of  its  arms ;  if  also  the  curve  VGG  is  such  that, 
when  any  straight  line  EFG  is  drawn  perpendicular  to  TD, 
cutting  the  lines  TD,  TH,  VGG  in  the  points  E,  F,  G,  and 
AF  is  joined,  EG  is  always  equal  to  AF;  then  the  line  VGG 
will  be  a  hyperbola. 


LECTURE    VI  75 

NOTE.  If  a  straight  line  FQ  is  drawn  perpendicular 
to  TH,  and  QR  is  taken  equal  to  AE  (along  TD),  and  GR 
is  joined;  then  GR  will  be  perpendicular  to  the  hyperbola 
VGG. 

Take  this  on  trust  from  me,  if  you  will,  or  work  it  out 
for  yourself;"*  I  will  waste  no  words  over  it. 

26.  Let  two  straight  lines,  AC,  BD,  intersecting  in  X,  be 
given  in  position;   then  if,  when  any  straight  line  PKL  is 
drawn  parallel  to  BA,  cutting  AC,  BD,  in  the  points  P  and 
K,  PL  is  always  equal  to  BK ;  then  the  line  ALL  will  be  a 
straight  line. 

27.  Let  a  straight  line  AX  be  given  in  position  and  a  fixed 
point  D;  also  let  the  line  DNN  be  such  that,  if  through  D 
any  straight  line  MN  is  drawn,  cutting  AX  in  M,  and  the  line 
DNN  in  N,  the  rectangle  contained  by  DM  and  DN  is  equal 
to  a  given  square,  say  the  square  on  Z;  then  the  line  DNN 
will  be  circular. 

Thus  you  will  see  that  not  only  a  straight  line  and  a 
hyperbola,  but  also  a  straight  line  and  a  circle,  each  in  its 
own  way,  are  reciprocal  lines  the  one  of  the  other. 

But  here,  although  we  have  not  yet  finished  our  preliminary 
theorems,  we  will  pause  for  a  while. 

NOTE 

It  has  already  been  noted  in  the  Introduction  that  the 
proofs  which  Barrow  gives  for  these  theorems,  even  in  the 
case  where  he  uses  an  algebraical  equation,  are  more  or  less 

*  "Ad  Calculum  exige." — I  hardly  think  that  Barrow  intends  "by 
analysis,"  but  he  may. 


76    BARROW'S  GEOMETRICAL   LECTURES 

of  a  strictly  geometrical  character;  the  terms  of  his  equa- 
tions are  kept  in  the  second  degree,  and  translated  into 
rectangles  to  finish  the  proofs.  In  this  connection,  note  the 
remark  on  page  197  to  the  effect  that  I  cannot  imagine 
Barrow  ever  using  a  geometrical  relation,  in  which  the  ex- 
pressions are  of  the  fourth  degree.  The  asymptotes  of  the 
hyperbolae  are  in  every  case  found ;  and  this  points  to  his 
intention  of  using  these  curves  as  auxiliary  curves  for  draw- 
ing tangents ;  cases  of  this  use  will  be  noted  as  we  come 
across  them ;  but  the  fact  that  the  number  of  cases  is  small 
suggests  that  the  paraboliforms,  which  he  uses  more 
frequently,  were  to  some  extent  the  outcome  of  his  re- 
searches rather  than  a  first  intention. 

The  great  point  to  notice,  however,  in  this  the  first  of  the 
originally  designed  seven  lectures,  is  that  the  idea  of  Time 
as  the  independent  variable,  i.e.  the  kinematical  nature  of 
his  hypotheses,  is  neglected  in  favour  of  either  a  geometrical 
or  an  algebraical  relation,  as  the  law  of  the  locus.  The  dis- 
tinction is  also  fairly  sharply  defined. 

Barrow,  throughout  the  theorems  of  this  lecture,  gives 
figures  for  the  particular  cases  that  correspond  to  rectangular, 
but  his  proofs  apply  to  oblique  axes  as  well  (in  that  he  does 
not  make  any  use  of  the  right  angle).  Both  the  figures  and 
the  proofs  have  been  omitted  in  order  to  save  space ;  all 
the  more  so,  as  this  lecture  has  hardly  any  direct  bearing 
on  the  infinitesimal  calculus. 

The  proofs  tend  to  show  that  Barrow  had  not  advanced 
very  far  in  Cartesian  analysis  ;  at  least  he  had  not  reached 
the  point  of  diagnosing  a  hyperbola  by  the  fact  that  the 
terms  of  the  second  degree  in  its  equation  have  real  factors ; 
or  perhaps  he  does  not  think  his  readers  will  be  acquainted 
with  this  method  of  obtaining  the  asympotes. 


LECTURE    VII 

Similar  or  analogous  curves.  Exponents  or  Indices. 
Arithmetical  and  Geometrical  Progressions.  Theorem  ana- 
logous to  the  approximation  to  the  Binomial  Theorem  for  a 
Fractional  Index.  Asymptotes. 

Barrow  opens  this  lecture  with  the  words,  "  Hitherto  we 
have  loitered  on  the  threshold,  nor  have  we  done  aught 
but  light  skirmishing."  The  theorems  which  follow,  as  he 
states  at  the  end  of  the  preceding  lecture,  are  still  of  the 
nature  of  preliminary  lemmas ;  but  one  of  them  especially, 
as  we  shall  see  later,  is  of  extraordinary  interest. 

For  the  rest,  it  is  necessary  to  give  some  explanation  of 
Barrow's  unusual  interpretation  of  certain  words  and  phrases, 
i.e.  an  interpretation  that  is  different  from  that  common  at  the 
present  time.  A  series  of  quantities  in  continued  proportion 
form  a  Geometrical  Progression  ;  thus,  if  we  have  A,  B,  C,  D  in 
continued  proportion,  then  these  are  in  Geometrical  Pro- 
gression, and  A:B=  B :  C  =  C  :  D.  Barrow  speaks  of  these 
as  being  "four  proportionals  geometrically,"  and  this 
accords  with  the  usual  idea.  But  he  also  speaks  of  "  four 
proportionals  arithmetically "  to  signify  the  four  quantities 
A,  B,  C,  D,  which  are  in  Arithmetical  Progression ;  that  is, 
A-B  =  B  -  Cj  =  C  -  D ;  and  further  his  proofs,  in  most  cases, 
only  demand  that  A  -  B  =  C  -  D.  If  A,  B,  C,  D,  E,  F,  .  .  .  N 
and  a,  b,  <r,  d,  e,  /  .  .  .  n  are  two  sets  of  proportionals, 
he  speaks  of  corresponding  terms  of  the  two  sets  as  being 
"of  the  same  order"  \  thus,  B,  b\  C,  c\  .  .  .  are  "mean 
proportionals  of  the  first,  second,  .  .  .  order  between  A  and 
N,  a  and  n  respectively;  and  this  applies  whether  the 
quantities  are  in  Arithmetical  or  Geometrical  Progression. 


78    BARROW'S  GEOMETRICAL  LECTURES 

An  index  or  exponent  is  also  defined  thus  : — If  the  number 
of  terms  from  the  first  term,  A  say,  to  any  other  term,  F 
say,  is  N  (excluding  the  first  term  in  the  count),  then  N 
is  the  index  or  exponent  of  the  term  F.  Later,  another 
meaning  is  attached  to  the  word  exponent-,  thus,  if  A,  B, 
C  are  the  general  ordinates  (or  the  radii  vectores)  of 
three  curves,  so  related  that  B  is  always  a  mean  of  the 
same  order,  say  the  third  out  of  six  means,  between  A  and 
C ;  so  that  the  indices  or  exponents  of  B  and  C  are  3  and 
7  respectively ;  then  3/7  is  called  the  exponent  of  the 
curve  BBB.  There  is  no  difficulty  in  recognizing  which 
meaning  is  intended,  as  Barrow  uses  njm  for  the  latter 
case,  instead  of  N/M.  The  connection  with  the  ordinary 
idea  of  indices  will  appear  in  the  note  to  §  1 6  of  this  lecture 
and  that  to  Lect.  IX,  §  4. 


1.  Let  A,  B  be  two  quantities,  of  which  A  is  the  greater  ;  let 
some  third  quantity  X  be  taken  ;  then  A  +  X:B  +  X<A:B. 

For,  since  X:A<X:B,  X  +  A:A<X  +  B:B;  hence,  etc. 

2.  Let  three  points,  L,  M,  N  be  taken  in  a  straight  line 

Y         L         E M       F  N    G      z 

Fig.  61. 

YZ ;  and  between  the  points  L,  M  let  any  point  E  be  taken, 
and  another  point  G  outside  LN  (towards  Z) ;  let  EG  be  cut 
in  F  so  that  GE  :  EF  =  NL  :  LM ;  then  F  will  fall  between 
M  and  Z. 

ForNE:ME  >  NL:ML(  =  GE:EF)>  NE:FE, 
.-.  FE  >  ME. 

3.  Let  BA,  DC  be  parallel  straight  lines,  and  also  BD,  GP; 
through    the    point    B    draw   two   straight   lines    BT,    BS, 


LECTURE    VII 


79 


cutting   GP   in    L  and    K ;   then  I  assert 
that  DS  :  DT  =  KG  :  LG 

For  the  ratio  KG  :  LG 
is  compounded  of  KG  :  GB  and  GB  :  LG, 
that  is,  of  PK  :  PS  and  PT  :  PL, 
that  is,  of  DB:DS  and  DT:DB, 
and  hence  is  equal  to  the  ratio  DT :  DS. 

4.  Let  BDT  be  a  triangle,  and  let  any  two  straight  lines 
BS,  BR,  drawn  through   B,  meet  any  straight  line  GP  drawn 
parallel  to  the  base  BD  in  the  points  L,  K ;  then  I  say  that 
LG.TD  +  KL.RD:KG.TD  =RD:8D. 


Fig.  62. 


[Barrow's  proof  by  drawing  parallels  through  L  is  rather 
long  and  complicated;  the  following  short  proof,  using  a 
different  subsidiary  construction,  is  therefore  substituted. 

Construction. — Draw   GXYZ,    as   in   the   figures,   parallel 

to   TD. 

Proof. —          Since   LG  :  BZ  =  GX  :  XZ  =  TS  :  SD, 
and   KG  :  BZ  =  GY  :  YZ  =  TR  :  RD; 
hence   RD  :  SD  =  LG  .  TR  :  KG  .  TS 

=  LG  .  TR  +  RD  .  KG  :   KG  .  TS  +  KG  .  SD 
=  LG  .  TD+  KL  .  RD  :  KG  .  TD.] 

5.  But,  if  the  points]  R,  8  are  not  situated  on  the  same 
side  of  the  point  D,*  then  by  a  similar  argument, 
LG.TD-KL.RD:  KG.TD=  RD  :  SD. 


*  This  is  a  mistake  :  D  should  be  P,  and  LG .  TD  -  KL .  RD  should  be 
ambiguous  in  sign. 


8o    BARROW'S  GEOMETRICAL  LECTURES 


6.  Let  there  be  four  equinumerable  series  of  quantities 
in  continued  proportion  (such  as  you  see  written  below), 
of  which  both  the  first  antecedents  and  the  last  conse- 
quents are  proportional  to  one  another  (i.e.  A  :  a  =  M  :  /*, 
and  F  :  <j>  =  8:0-);  then,  the  four  in  any  the  same  column 
being  taken,  they  will  also  be  proportional  to  one  another 
(say,  for  instance,  D  :  8=  P  :  TT). 

A,     B,    C,     D,    E,    F 

a,    A  y>    8,    *,    <£ 
M,    N,    0,    P,    R,    8 


For  A/x,   BV,    Co,    DTT,  E/o,   Fa-    i 

M    OKI      n   *D      D     .0  f  are  in  continued  proportion. 
and      aM,  /3N,  yO,  8P,  eR,  <£S  J 

Therefore,  since  A/u  =  Ma  and  Fa-  =  <£S,  it  is  plain  that 
DTT  =  SP,  and  hence  that  D  :  8  =  P  :  TT. 

The  conclusion  applies  equally  to  either  Arithmetical  or 
Geometrical  proportionality.* 

7.  Let  AB,  CD  be  parallel  lines; 
and  let  a  straight  line  BD,  given 
in  position,  cut  these.  Now  let 
the  curves  EBE,  FBF  be  so  related 
that,  if  any  straight  line  PG  is 
drawn  parallel  to  BD,  PF  is  always 
a  mean  proportional  of  the  same  T  Fig.  65. 

*  If  the  series  are  "arithmetical  proportionals,"  then 

*#.  S:S,?JS,  °i5;  f+ff.  £S  H  arithmetical  proportions  ; 

and  the  condition  for  the  first  antecedents  and  the  last  consequents  must  be 
A  +/j.=  a+M  and  F  +  er  =  <f>  +  S  ;  in  this  case  D  +  TT  =  £+P.  or  D-8  =  P-  ?r. 
This  is  not  of  very  great  importance,  as,  in  the  following  theorems, 
Barrow  apparently  only  considers  geometrical  proportionals.  For  arith- 
metical proportionality,  the  lines  AGB,  HEL  must  be  parallel  curves,  i.e. 
EG  must  be  constant,  and  then  the  curves  KEK  and  FBF  are  parallel 
curves,  i.e.  RK  is  constant. 


LECTURE    VII  8 1 

order  between  PG  and  PE;  then,  through  any  point  E  of 
the  given  line  EBE,  let  HE  be  drawn  parallel  to  AB  and  CD ; 
and  let  another  curve  KEK  be  such  that,  if  any  straight  line 
QL  is  drawn  also  parallel  to  BD,  cutting  EBE  in  I,  HE  in  L, 
FBF  in  R,  and  CD  in  8,  then  QK  is  always  a  mean  propor- 
tional of  the  same  order  between  QL  and  Ql  (of  that  order, 
I  say,  of  which  PF  was  a  mean  proportional  between  PG  and 
PE) ;  then  I  assert  that  the  lines  FBF,  KEK  are  similar  ;  that 
is,  the  ordinates  such  as  QR,  QK  bear  a  constant  ratio  to 
one  another,  the  ratio  which  PF  bears  to  PE. 

This  follows  from  the  lemma  just  proved,  as  will  be  clear 
by  considering  the  argument  below. 


Since  QS,  QR,  Ql 
QL,  QK,  Ql 
PG,  PF,  PE 
PE,  PE,  PE 


are  proportionals 

such  that 

QS:QL=  PG:PE, 
Ql  :QI    =  PE:PE; 


hence,  QR  :  QK  =  PF  :  PE 

NOTE.     Instead  of  the  straight  lines  AB,  HE,  CD,  we  can 

substitute  any  parallels  we  please,  even  curved  lines. 

8.  Again,  let  AQPB,  A8GD  be  two  straight  lines  meeting  in 
A,  and  let  BD  be  a  straight  line  given  in  position ;  also  let 
EBE,  FBF  be  two  curves  so  related  that,  if  any  straight  line 
PG  is  drawn  parallel  to  BD,  PF  is  always  a  mean  propor- 
tional of  the  same  order  between  PG  and  PE ;  then,  having 
joined  AE,  let  another  curve  KEK  be  such  that,  if  any 
straight  line  QLI  is  drawn  parallel  to  BD,  cutting  AE  in  L, 
EBI  in  I,  and  FBF  in  R,  QK  is  always  a  mean  proportional 

6 


82    BARROW'S  GEOMETRICAL  LECTURES 

between  QL  and  Ql  of  the  same  order  as  PF  was  between 
PG  and  PE;  then  the  line  FBF  is  similar  to  the  line  KEK; 
that  is,  QR  :  QK  =  PF,:  PE,  in  every  position. 

NOTE.     For  the  three  straight  lines  A B,  AH,  AD  we  can 
substitute  any  three  analogous  lines. 

9.  Also,  if  AGB  is  a  circle  whose  centre  is  D  ;  and  EBE, 
FBF  are  two  other  curves  such  that,  if  any  straight  line  DG 
is  drawn  through  D,  DF  is  always  a  mean  proportional  of 
the  same  order  between  DG  and  DE ;  then,  through  E,  let  a 
circle  H  E,  with  centre  D,  be  drawn  ;  and  let '  another  curve 
KEK  be  drawn  such  that,  if  any  straight  line  DL  is  drawn 
through  D  to  meet  the  circle  HE  in  L,  and  EBE  in  I,  DK  is 
always  a  mean  proportional  between  DL  and  Dl,  of  the  same 
order   as   DF  was  between   DG   and  DE;    then  the  curves 
FBF,  KEK  will  be  similar,  i.e.  DR  :  DK  =  DF  :  DE,  in  every 
position.     [DL  meets  FBF  in  R.] 

NOTE,     in  this  case  also,  instead  of  the  circles,  we  may 
substitute  any  two  parallel  or  two  analogous  lines. 

10.  Lastly,  let  AGBG,  EBE  be  any  two  lines;  and  let  FBF 
be  another  line  so  related  to  them  that,  if  any  straight  line 
DG  be  drawn  in  any  manner  through  a  fixed  point  D,  DF  is 
always  a  mean  proportional  of  the  same  order  between  DG 
and  DE;  then  let  H  EL  be  a  line  analogous  to  AGB  (i.e.  such 
that,  if  through    D    a   straight  line  DSL  is  drawn  in   any 
manner,  DS  and  DL  are  always  in  the  same  proportion); 
lastly,  let  the  line  KEK  be  such  that,  if  DL  be  drawn  in  any 
manner,  cutting  EBE  in  I,  DK  is  always  a  mean  proportional 
between  DL  and  Dl,  of  the  same  order  as  DF  was  between 


LECTURED  VII  83 

DG  and  DE ;  then,  in  this  case  also,  FBF  is  analogous  to  the 
line  KEK. 

11.  Let  A,  B,  C,  D,  E,  F  be  a  series  of  quantities  in  Arith- 
metical Progression ;  and,  two  terms  D,  F  being  taken  in  it, 
let  the  number  of  terms  from  A  to  D  (excluding  A)  be  N, 
and  the  number  of  terms  from  A  to  F  (excluding  A)  be  M  ; 
then  A~D:A~F  =  N  :  M. 

For,  suppose  the  common  difference  to  be  X ;  then 

D  =  A  ±  N  .  X     and     F  =  A  ±  M  .  X ; 
therefore         A~D:A~F  =  N  .  X  :  M  .  X  =  N  :  M. 

12.  Hence,  if  there  are  two  series  of  this  kind,  and  in 
each  a  pair  of  terms,  corresponding  to  one  another  in  order, 
are  taken  (say  D,  F  in  the  first,  and  P,  R  in  the  second) ; 
then  A~D:A~F  =  M~P:M~R, 

where  the  series  are 

A,  B,  C,  D,  E,  F    and     M,  N,  0,  P,  Q,  R. 
For   each    of  these   ratios   is    equal    to  that   which  the 
numbers,  N,  M,  as  found  in  the  preceding  article,  bear  to 
one  another. 

These  numbers  N,  M,  in  any  series  of  proportionals,  we 
shall  usually  call  the  exponents  or  indices  of  the  terms  to 
which  they  apply ;  and  where  we  use  these  letters  in  what 
follows,  we  shall  always  understand  them  to  have  this 
meaning. 

13.  Let  any  quantities   A,   B,  C,   D,   E,   F  be  a  series  in 
Arithmetical   Progression ;   and  let   there   be   another   set, 
equal    in    number,    in    Geometrical    Progression,    starting 


84   BARROW'S  GEOMETRICAL  LECTURES 

with   the   same   term    A  ;    thus,    [suppose   the   two    series 

are] 

A,  B,  C,  D,  E,  F. 

A,  M,  N,  0,  P,  Q. 

Also  let  the  second  term  B  of  the  Arithmetical  Progression 
be  not  greater  than  M  the  second  term  of  the  Geometrical 
Progression ;  then  any  term  in  the  Geometrical  Progression 
is  greater  than  the  term  in  the  Arithmetical  Progression 
that  corresponds  to  it.* 

For,  A  +  N  >  2M  >  2B  or  A  +  C,  .'.  N  >  C; 

hence,     M  +  N  >  B  +  CorA  +  D;  butA  +  0  >  M-fN; 

.%  A 4-0  >  A  +  D,     i.e.     0  >  D; 
hence,     M  +  0  >  B  +  DorA  +  E;  butA  +  P>  M+0; 

.'.  A+P  >  A  +  E,     i.e.     P  >  E; 
and  so  on,  as  far  as  we  please. 

14.  Hence,  if  A,  B,  C,  D,  E,  F  are  in  Arithmetical  Pro- 
gression, and  A,  M,  N,  0,  P,  Q  are  in  Geometrical  Progression, 
and  the  last  term   F  is  not  less  than  the  last  term  Q  (the 
number  of  terms  in  the  two  series  being  equal) ;  then  B  is 
greater  than  M. 

For,  if  we  say  that  B  is  not  greater  than  M,  then  F  must 
be  less  than  Q ;  which  is  contrary  to  the  hypothesis. 

15.  Also,  with  the  same  data,  the  penultimate  E  is  greater 
than  the  penultimate  P. 

*  Algebraically:— If  the  series  are  a,  a  +  d,  a  +  2d,  a  +  ^d,  .  .  .  and 
a,  ar,  at*,  ar^,  .  .  .  ,  we  have,  whether  r  ^  i,  the  fact  that  (i-r)(i-r), 
(i  — r)(i  — r2),  (i  —  r)(i  — r3),  .  .  .  are  all  positive  ;  hence  it  follows  that 

a  +  ar2  >  zar,     a  +  ar3  >  ar+ar2,     a  +  ar4  >  ar  +  ar3,  .  .  . 

Hence,  since  ar  is  not  less  than  a-\-d,  it  follows  that 

a  +  arz  >  z(a  +  d)  or     «/"2> 

a  +  ar8  >  (a  +  d)  +  (a  +  2d)     or     at^> 

and  so  on,  in  exact  equivalence  with  Barrow's  proof. 


LECTURE    VII  85 

1 6.  Moreover,  with  the  same  data,  any  term  in  the  Arith- 
metical series  is  greater  than  any  term  in  the  Geometrical 
series  ;  for  instance,  C  >  N. 

For  E  >  P,  and  hence  D  >  0,  and  so  on,  .-.  C  >  N.* 

17.  Hence  it  may  be  proved  that : — If  there  are  any  four 
lines  HBH,  GBG,  FBF,  EBE,  cutting  one  another  at  B,  and 
these  are  so  related  that,  if  any  straight  line  DH  is  drawn  in 
any  manner  parallel  to  a  straight  line  BD,  given  in  position, 
or  if  through  a  given  point  D  any  straight  line  DH  is  drawn, 
DG  is  always  an  arithmetical  mean  of  the  same  order  be- 
tween DH  and  DE,  and  DF  is  the  geometrical  mean  of  the 
same  order ;  then  the  lines  GBG,  FBF  will  touch  one  another. 
For  it  is  evident  from  the  preceding  that  the  line  GBG  will 
lie  wholly  outside  the  line  FBF. 

1 8.  Hence  also   (I    mention   it   briefly  in   passing),  the 
asymptotes,    straight     lines    applying    to    many    different 
kinds  of  hyperbolae,  and  curves  of  hyperbolic  form,  may 
be  defined. t 

Thus,  let  two  straight  lines  VD,  BD  be  given  in  position, 
and  let  AGB,  VEI  be  two  other  straight  lines;  now,  any 
straight  line  PG  being  drawn  parallel  to  DB,  let  P</>  be 
always  an  arithmetical  mean  of  the  same  order  between 
PG  and  PE,  and  let  PF  be  the  geometrical  mean  of  the 
same  order.  Now,  since  the  straight  lines  EG,  E<£  are 
always  in  the  same  ratio,  the  line  <£<£<£  is  a  straight  line; 
but  the  line  VFF  is  a  hyperbola  or  some  curve  of  hyperbolic 

*  See  note  at  the  end  of  this  lecture  ;  the  italics  are  mine, 
f  See  note  at  the  end  of  the  lecture. 


86   BARROW'S   GEOMETRICAL  LECTURES 

form  (the  hyperbola  of  Apollonius  indeed,  if  PF  is  the 
simple  geometrical  mean  between  PG  and  PE,  but  some 
curve  of  hyperbolic  form  of  a  different  kind,  if  PF  is  a  mean 
of  some  other  kind) ;  and  it  is  plain  from  the  last  theorem 
that  the  line  <£<£<£  is  an  asymptote  to  the  line  VFF,  corre 
sponding  to  the  points  of  the  same  kind  of  mean. 

I  do  not  know  whether  this  is  of  very  much  use,  but 
indeed  it  was  an  incidental  corollary  for  us  to  have  obtained 
it  here. 

19.  Let  three  straight  lines  BA,  BC,  BQ  be  drawn  through 
a  given  fixed  point  B  to  a  straight  line  AC,  fixed  in  position ; 
then,  in  QC  produced  let  some  point  D  be  taken  as  a  fixed 
point.     Then    it  is  possible  to  draw  through  B  a  straight 
line  (BR  say),  on  either  side  of  BQ,  such  that,  if  any  straight 
line  is  drawn  through  D,  as  DN,  the  part  intercepted  between 
BQ  and  BR  is  less  than  the  part  intercepted  between  BA 
and  BC. 

20.  Let  D,  E,  F  be  three  points  in  a  straight  line  DZ,  and 
let  F  be  the  vertex  of  a  rectilineal  angle  BFC,  of  which  the 

-arms  are  cut  by  a  straight  line  DBG;  let  a  straight  line  EG 
be  drawn  through  E ;  then  it  is  possible  to  draw  through 
E  a  straight  line  EH  such  that,  on  any  straight  line  DK 
drawn  through  the  point  D,  the  intercept  between  the 
lines  EG,  EH  is  less  than  the  intercept  between  the  lines 
FC,  FB. 

21.  Let  a  straight  line  BO  touch  a  curve  BA  in  B;  and 
let  the  length  BO  of  the  straight  line  be  equal  to  the  arc 
BA  of  the  curve ;  then,  if  any  point  K  is  taken  in  the  arc 


LECTURE    VII  87 

BA  and  KO  is  joined,  the  straight  line  KO  is  greater  than 
the  arc  KA. 

22.  Hence,  if  any  two  points  K,  L,  on  the  same  side  of 
the  point  of  contact,  are  taken,  one  in  the  curve  and  the 
other  in  the  tangent,  and  KL  is  joined;  then  KL-fLO  > 
arc  KA. 


NOTE 

From  §  1 6  we  have  the  following  geometrical  theorem. 

Suppose  a  line  AB  is  divided  into  two  parts  at  C,  and 
that  the  part  CB  is  divided  at  D,  E,  F,  G,  H  in  the  figure  on 
the  left,  and  at  D',  E',  F',  G',  H'  in  the  figure  on  the  right, 
so  that  AC,  AD,  AE,  AF,  AG,  AH,  AB  are  in  Arithmetical  Pro- 
gression, and  AC,  AD',  AE',  AF',  AG',  AH',  AB  are  in  Geo- 
metrical Progression;  then  AD  >  AD',  .  .  .,  AH  >  AH'. 

A CDEFGHE*  A dd/fe^'d1  H1  '  B 

Expressing  this  theorem  algebraically,  we  see  that,  if 
AC  =  a  and  CB  =  ax,  and  the  number  of  points  of  section 
between  C  and  B  is  n—  i,  and  F  is  the  rt\\  arithmetical, 
and  F'  the  rth  geometrical  "  mean  point "  between  C  and 
B,  then  the  relation  AF  >  AF'  becomes 

a  +  r.  ax/n  >  a  .  [  l/{(a  +  ax)/a}]r; 
i.e.  i  +  (r/n)x  >  (i  +x)r/n,     where     r  <  n. 

Also,  as  CB  becomes  smaller  and  smaller,  the  difference  FF 
becomes  smaller  and  smaller,  since  it  is  clearly  less  than 
CB ;  that  is,  the  ratio  FF'/AC  can  be  made  less  than  any 
assigned  number  by  taking  the  ratio  CB/AC  small  enough. 
Hence  the  algebraical  inequality  tends  to  an  equality,  when 
x  is  taken  smaller  and  smaller. 

Again,  if  we  put  rx/n  =  y,  we  have  x  =  ny/r,  and  then 

i  +y  >  ( i  +  ny/rY/n     or     i  +  (n/r)y  <  ( i  +y)n/r, 
where  n  >  r\  and  again  the  inequality  tends  to  become  an 
equality  \S  y  is  taken  small  enough. 


88    BARROW'S  GEOMETRICAL  LECTURES 


Naturally,  a  man  who  uses  the  notation  xx  for  x2  does 
not  state  such  a  theorem  about  fractional  indices.  But  the 
approximation  to  the  binomial  expansion  is  there  just  the 
same,  though  concealed  under  a  geometrical  form.  We 
may  as  well  say  that  the  ancient  geometers  did  not  know 
the  expansion  for  sin  (A  +  B),  when  they  used  it  in  the  form 
of  Ptolemy's  theorem,  as  say  that  Barrow  was  unaware  of 
this.  Moreover,  if  further  corroborative  evidence  is  needed, 
we  have  it  in  §  18.  Here  Barrow  states  that  a  line  </><£</>  is 
an  asymptote  to  a  curve  VFF,  the  distance  between  the  curve 
and  its  asymptote,  measured  along  a  line  parallel  to  a  fixed 
direction,  being  the  equivalent  of  our  FF'  in  the  work  above. 
Now  the  condition  for  an  asymptote  is  that  this  distance 
should  continually  decrease  and  finally  become  evanescent 
as  we  proceed  to  "infinity."  Let  us  try  to  reason  out  the 
manner  in  which  Barrow  came  to  the  conclusion  that  his 
line  was  an  asymptote  to  his  curve. 


The  figure  on  the  left  is  the  one  used  by  Barrow  for  §  18; 
as  P  moves  away  from  V,  PE  and  PG  both  increase  without 
limit,  but  it  can  readily  be  seen  that  the  ratio  of  EG  to  PE 
steadily  decreases.  This  is  all  that  can  be  gathered  from 
the  figure ;  and,  as  far  as  I  can  see,  it  must  have  been  from 
this  that  Barrow  argued  that  the  distance  F</>  decreased 
without  limit  and  ultimately  became  evanescent.  In  other 
words,  he  appreciated  the  fact  that  the  inequality  tended  to 
become  an  equality  when  x  was  taken  small  enough.  Assum- 
ing that  my  suggestion  is  correct,  the  very  fact  that  he  has 
recognized  this  important  truth  leads  him  into  a  trap ;  for 
the  line  <f><f><j>  is  not  an  asymptote  to  the  curve  VFF,  i.e.  as  we 
understand  an  asymptote  at  the  present  day.  Taking  the 


LECTURE    VII  89 

simplest  case,  as  mentioned  by  Barrow,  of  the  ordinary 
hyperbola,  it  is  readily  seen  that  the  other  branch  of  the 
curve  passes  through  the  common  point  of  the  straight  lines 
AGB,  VEI,  and  therefore  the  line  <£<£<£  cannot  bean  asymptote, 
for  it  also  passes  through  this  common  point  and  touches 
the  curve  there. 

This  is  easily  seen  analytically,  taking  the  figure  on  the 
right.  For,  if  the  equations  of  VEI  and  AGB,  referred  to 
VD  and  a  line  parallel  to  DB  through  the  middle  point  of 
AV  as  axes,  are  _y  =  n(x  +  a)  and  y  =  m(x  -  a),  then  the  equa- 
tion to  the  hyperbola  isjy2  =  mn(x2  -  a2) ;  that  of  theasymptote, 
with  which  Barrow  confuses  the  line  ^><£<£,  isy  =  ,J(mri) .  x\ 
and  that  of  the  line  <£<£<£  is  2y  =  (m  +  n)x  +  (m-  ri)a  •  and 
the  two  lines  are  not  the  same  unless  m  =  n,  i.e.  unless  VEI 
and  AGB  are  parallel.  The  argument  is  the  same,  if  DB  is 
not  taken  at  right  angles  to  VD,  or  for  different  kinds  of 
"  means." 

The  true  source  of  the  error  is,  of  course,  that  it  is  not 
true  that  F</>  decreases  without  limit,  but  that  it  is  F<£  :  PE 
which  decreases  without  limit,  whilst  PE  increases  without 
limit.  This  kind  of  difficulty  is  exactly  on  a  par  with  the 
difficulties  arising  from  considerations  of  convergence  of 
infinite  series.  Barrow  certainly  has  in  his  theorem  the 
equivalent  of  the  binomial  approximation  as  far  as  it  is 
necessary  for  differentiation  of  fractional  powers  in  the 
ordinary  method ;  it  is  very  likely  that  he  may  have  found 
difficulties  with  other  theorems  of  the  kind  discussed  above ; 
but,  as  will  be  seen  in  the  note  to  Lect.  IX,  §  4,  he  is  quite 
independent  of  considerations  of  this  sort,  i.e.  of  infinite 
series  with  all  their  difficulties ;  for  all  that  he  requires  is 
the  bare  inequality,  as  given  in  his  theorem.  By  means  of 
this,  at  the  very  least,  he  was  the  first  man  to  give  a  rigorous 
demonstration  of  a  method  for  differentiating  a  fractional 
power  of  the  variable. 

As  an  example  of  the  use  that  a  geometer  could  make  of 
his  geometrical  facts,  it  may  be  pointed  out  that  the  theorem 
of  §  17  is  equivalent  to  the  analytical  theorem  : — 

The  curves  whose  equations  are 

y  =  [{«  -  '}  .f(x)  +  r .  F(x)]ln  and  y  =  ?/[{/(*)}"-• .  {/(*)}•] 
touch  one  another  at  all  the  points  common  to  the  two 
curves  whose  equations  are  y  =  f(x)  and  y  =  F(x). 


LECTURE   VIII 

Construction  of  tangents  by  means  of  auxiliary  curves  of 
which  the  tangents  are  known.  Differentiation  of  a  sum  or  a 
difference.  Analytical  equivalents. 

Truly  I  seem  to  myself  (and  perhaps  also  to  you)  to  have 
done  what  that  wise  man,  the  Scoffer,*  ridiculed,  namely, 
to  have  built  very  large  gates  to  a  very  small  city.  For  up 
to  the  present,  we  have  dqjie  nothing  else  but  struggle 
towards  the  real  thing,  just  a  little  nearer.  Now  let  us  get 
to  it. 

1.  We  assume  the  following : — 

If  two  lines  OMO,  TMT  touch  one  another,  the  angles 
between  them  (OMT)  are  less  than  any  rectilineal  angle; 
and  conversely,  if  two  lines  contain  angles  which  are  less 
than  any  rectilineal  angle,  they  touch  one  another  (or  at 
least,  they  will  be  equivalent  to  lines  that  touch). 

The  reason  for  this  statement  has  already  been  discussed, 
unless  I  am  mistaken. 

2.  Hence,  if  any  third  line  PMP  touch  two  lines  OMO, 
TMT,  the  lines  OMO,  TMT  will  also  touch  one  another. 

*  Socrates,  the  Athenian  philosopher  :  Zeno  called  him  "  Scurra  Atticus." 
the  Attic  Scoffer. 


LECTURE    VIII  91 

3.  Let  a  straight  line  FA  touch  a  curve  FX  in  F ;  and  let 
FE  be  a  straight  line  given  in  position ;  also  let  EY,  EZ  be 
two  curves  such  that,  if  any  straight  line  IL  is  drawn  parallel 
to  EF,  cutting  FA  in  G  and  the  curves  FX,  EY,  EZ  in  I,  K,  L 
respectively,  the  intercept  KL  is  always  equal  to  the  intercept 
IG  ;  then  the  curves  EY,  EZ  touch  one  another. 

4.  Again,  let  a  straight  line  AF  touch  a  curve  AX,  and  let 
EY,  EZ  be  two  curves  such  that,  if  through  a  fixed  point  D 
any  straight  line  DL  is  drawn,  cutting  the  given  lines  as  in 
the  preceding  theorem,  KL  is  always  equal  to  IG  ;  then  the 
curves  EY,  EZ  will  touch  one  another. 

The  two  foregoing  conclusions  are  also  true,  and  can 
be  shown  to  be  true  by  a  like  reasoning,  if  it  is  given 
that  IG,  KL  always  bear  to  one  another  any  the  same 
ratio. 

5.  Let  TEI  be  a  straight  line,  and  let  two  curves  YFN,  ZGO 
be  so  related  that,  if  any  straight  line  EFG  is  drawn  parallel 
to  AB,  a  straight  line  given  in  position, 

the  intercepts  EG,  EF  always  bear  to 
one  another  the  same  ratio  ;  also  let 
the  straight  line  TG  touch  ZGO,  one  of 
the  curves  in  G  and  meet  IE  in  T; 
then  TF,  being  joined,  will  touch  *the 
curve  YFN.  Fig.  80. 

For,  let  a  straight  line  IL,  parallel  to  AB,  be  drawn,  cutting 
the  given  lines  as  shown  in  the  figure.  Then 

IL:IN  >  10:  IN  >  EG:EF  >  IL:  IK,  and  .-.  IN  <  IK; 


92    BARROW'S  GEOMETRICAL  LECTURES 

hence  the  line  TF  falls  altogether  without  the  curve  YFN.* 

Otherwise.  It  can  be  shown  that  IL:IK  =  OL:NK;  hence, 
by  §  4  above,  since  GL,  GO  touch,  FN,  FK  also  touch. 

6.  Moreover,  if  three  curves  XEM,  YFN,  ZGO  are  so  related 
that,  if  any  straight  line  EFG  is  drawn  parallel  to  a  line  given 
in  position,  EG  and  EF  are  always  in  the  same  ratio  ;  also 
let  the  tangents  ET,  GT  to  the  curves  XEM,  ZGO  meet  in  T; 
then  TF,  being  joined,  will  touch  the  curve  YFN.t 

7.  Let   D  be  a  given    point,  and  let  XEM,  YFN   be  two 
curves  so  related  that,  if  through  D  any  straight  line  DEF  is 
drawn,  the  straight  lines  DE,  DF  always  bear  to  one  another 
the  same  ratio;  and  let  the  straight  line  FS  touch  YFN,  one 
of  the  curves,  and  let  ER  be  parallel  to  FS;  then  ER  will 
touch  the  curve  XEM.J 

8.  Let  XEM,  YFN,  ZGO  be  three  curves  such  that,  if  any 
straight  line  DEFG  is  drawn  through  a  given  point  D,  the 

*  The  reasoning  for  these  theorems  given  by  Barrow  is  not  conclusive  ; 
it  depends  too  much  on  the  accident  of  the  figure  drawn.  Although  he 
states  in  a  note  after  §  6  that  he  always  chooses  the  simplest  cases,  it  is 
desirable  that  these  simple  cases  should  be  capable  of  being  generalized 
without  altering  the  argument.  In  addition,  his  proof  of  §  4  is  long  and 
complicated,  and  necessitates  as  a  preliminary  lemma  the  theorem  of 
Lect.  VII,  §  20;  this  is  also  proved  in  a  far  from  simple  manner,  although 
there  is  a  very  simple  proof  of  it.  Still  Barrow  must  have  had  some  good 
reason  for  proving  these  two  theorems  by  the  method  of  "  the  vanishing 
angle"  of  §  i,  for  he  states  that  "these  theorems  are  set  forth,  so  that  none 
of  the  following  theorems  maybe  hampered  with  doubts."  He  seems  to 
doubt  the  rigour  of  the  method  used  in  §  5,  of  which  I  have  given  the  full 
proof  for  the  sake  of  exemplification  ;  together  with  the  alternative  proof 
by  §  4.  The  proof  of  the  latter  follows  easily  thus  : — Since  FA  lies  wholly 
on  one  side  of  the  curve  FX,  EZ  lies  wholly  on  one  side  of  EY. 

f  If  y  =/(x),  y  =  F(.r),  y  =  $(x}  are  three  curves  such  that  there  is  a  con- 
stant relation  A./+B.  F  +  C.<£  =  o,  where  A  +  B  +  C  =  o,  the  tangents  at 
points  having  equal  abscissas  are  concurrent. 

J  Homothetic  curves  have  parallel  tangents  ;  this  theorem  and  the  next 
are  the  polar  equivalents  of  those  of  §§  5,  6. 


LECTURE    VIII  93 

intercepts  EG,  EF  always  bear  the  same  ratio  to  one  another 
(say  as  R  is  to  S) ;  and  let  the  straight  lines  ET,  GT  touch 
two  of  the  curves  (say  XEM,  YFN)  in  E  arid  G  ;  it  is  required 
to  draw  the  tangent  at  F  to  the  curve  YFN. 

Imagine  a  curve  TFV  such  that,  if  a  straight  line  is  drawn 
in  any  manner  through  D,  cutting  the  straight  lines  TE,  TG 
in  the  points  I,  L  and  the  curve  in  K,  the  intercepts  IL,  IK 
bear  to  one  another  the  same  given  ratio,  R  to  8.  Then 
IK  >  IN,  and  therefore  the  curve  TFK  touches  the  curve  YKN. 
But,  by  Lect.  VI,  §  4,  the  curve  TFK  is  a  hyperbola;*  let 
FS  be  the  tangent  to  it.  Then  SF  will  touch  the  curve 
YFN  also. 

Since  mention  is  here  made  for  the  first  time  of  a  tangent 
to  a  hyperbola,  we  will  determine  the  tangent  to  this  curve, 
together  with  the  tangents  of  all  other  curves  constructed 
by  a  similar  method,  or  of  reciprocal  lines. 

9.  Let  VD  be  a  straight  line,  and  XEM,  YFN  two  curves 
so  related  that,  if  any  straight  line  EOF  is  drawn  parallel  to 


Fig.  84. 

a  line  given  in  position,  the  rectangle  contained  by  DE,  DF 
is  always  equal  to  any  the  same  area  ;  also  the  straight  line 

*  Note  the  use  of  the  auxiliary  hyperbola. 


94   BARROW'S  GEOMETRICAL  LECTURES 

ET  touches  the  curve  XEM  at  E,  and  cuts  YD  in  T  ;  then,  if 
DS  is  taken  equal  to  DT  and  FS  is  joined,  FS  will  touch  the 
curve  YFN  at  F.* 


Let  any  straight  line  IN  be  drawn  parallel  to  EF,  cutting 
the  given  lines  as  shown  ;  then 

TP  :  PM  >  TP  :  PI  >  TD  :  DE ;  also  SP  :  PK  =  DS  :  DF  ; 
TP .  SP  :  PM  .  PK  >  TD  .  8D  :  DE .  DF  >  TD .  SD  :  PM  .  PN. 
But,  since  D  is  the  middle  point  of  TS, .?.  TD .  SD  >  TP  .  SP 
hence  all  the  more,  TD  .  SD  :  PM  .  PK  >  TD .  SD  :  PM  .  PN, 

/.  PM.PK<  PM.PN    or    PK  <  PN. 
Therefore  the  whole  line  FS  lies  outside  the  curve  YFN. 

NOTE.  If  the  line  XEM  is  a  straight  line,  and  so  coinci- 
dent with  TEI,  the  curve  YFN  is  the  ordinary  hyperbola,  of 
which  the  centre  is  T  and  the  asymptotes  are  TS  and  a  line 
through  T  that  is  parallel  to  EF. 

10.  Again,  let  D  be  a  point,  and  XEM,  YFN  two  curves  so 
related  that,  if  any  straight  line  EF  is  drawn  through  D,  the 
rectangle  contained  by  DE,  DF  is  always  equal  to  a  certain 

*  The  analytical  equivalent  of  this  is  : — 

If  y  is  a  function  of  x,  and  z  =  A//,  then  (i/z) .  dzfdx  —  -  (i/y) .  dyjdx. 
Also  the  special  case  gives  d(i/x)/dx  =  -  i/x2.  It  is  thus  that  Barrow  starts 
his  real  work  on  the  differential  calculus. 


LECTURE    VIII  95 

square  (say  the  square  on  Z)  ;  and  let  a  straight  line  ER 
touch  one  curve  XEM;  then  the  tangent  to  the  other  is 
found  thus: — 

Draw  DP  perpendicular  to  ER  and,  having  made  DP  :  Z  = 
Z  :  DB,  bisect  DB  at  C  ;  join  CF  and  draw  FS  at  right  angles 
to  CF;  then  FS  will  touch  the  curve  YFN, 

11.  Let  XEM  and  YFN  be  two  curves  such  that,  if  any 
straight  line  FE  is  drawn  parallel  to  a  straight  line  given  in 
position,  it  is   always  equal  to  a  given  length ;  also  let  a 
straight  line  FS  touch  the  curve  YFN  ;  then  RE,  being  drawn 
parallel  to  FS,  will  touch  the  curve  XEM.* 

12.  Let  XEM   be  any  curve,   which  a  straight  line  ER 
touches  at  E ;  also  let  YFN  be  another  curve  so  related  to 
the  former  that,  if  a  straight  line  DEF  is  drawn  in  any  manner 
through  a  given  point  D,  the  intercept  EF  is  always  equal 
to  some  given  length  Z ;  then  the  tangent  to  this  curve  is 
drawn  thus : — 

Take  DH  =  Z  (along  DEF),  and  through  H  draw  AH 
perpendicular  to  DH,  meeting  ER  in  B;  through  F  draw  FG 
parallel  to  AB;  take  GL  =  GB  ;  then  LFS,  being  drawn, 
will  touch  the  curve  YFN.t 

NOTE.  If  XEM  is  supposed  to  be  a  straight  line,  and  so 
coincide  with  ER,  then  YFN  is  the  ordinary  true  Conchoid, 
or  the  Conchoid  of  Nicomedes ;  hence  the  tangent  to  this 
curve  has  been  determined  by  a  certain  general  reasoning. 

*  The  analytical  equivalent  is  : — If  y  is  a  function  of  x,  and  z  =  y+c, 
where  c  is  a  constant,  then  dzjdx  =  dyjdx. 

f  For  the  proof  of  this  theorem,  Barrow  again  uses  an  auxiliary  curve, 
namely  the  hyperbola  determined  in  Lect.  VI,  §  9. 


96    BARROW'S   GEOMETRICAL  LECTURES 

13.  Let  VA  be  a  straight  line,  and  BEI  any  curve  ;  and  let 
DFG  be  another  line  such  that,  if  any  straight  line  PFE  is 
drawn  parallel  to  a  line  given  in  position,  the  square  on 
PE  is  equal  to  the  square  on  PF  with  the  square  on  a  given 
straight  line  Z  ;  also  let  the  straight  line  TE  touch  the  curve 
DEI;  let  PE2:PF2  =  PT:PS;  then  FS,  being  joined,  will 
touch  the  curve 


[This  is  proved  by  the  use  of  Lect.  VI,  §  22,  and 
corollary.] 

14.  Let  other  things  be  supposed  the  same,  but  now  let 
the  square  on  PE  together  with  the  square  on  Z  be  equal 
to  the  square  on  PF;  also  let  PE2:  PF2  =  PT:  PS  ;  then  FS 
will  touch  the  curve  GFG.f 

[For  this,  Barrow  uses  Lect.  VI,  §  23,  and  its  corollary.] 

15.  Let  AFB,  CGD  be  two  curves  having  a  common  axis 
AD,  so  related  to  one  another  that,  if  any  straight  line  EEG 
is  drawn  perpendicular  to  AD,  cutting  the  lines  drawn   as 
shown,  the  sum  of  the  squares  on  EF  and  EG  is  equal  to 
the  square  on  a  given  straight  line  Z;  also  let  the  straight 
line   FR  touch  AFB,  one  of  the  curves;  and  let  EF2  :  EG2 

=  ER  :  ET  ;  then  GT,  being  joined,  will  also  touch  the  curve 
CGD.  I 

[For  this,  Barrow  uses  Lect.  VI,  §  24,  and  its  corollary.] 

*  The  analytical  equivalent  is  :  —  If  y  is  any  function  of  *•,  and  22=  _y2  —  a2, 
where  a  is  some  constant,  then  z.dz/dx=  y.dyjdx  ;  or  in  a  different  form, 
if  z=  V(v2  -  a2),  then  dzjdx=y.  (dy/dx)/(yz  -  a2).  The-  particular  case,  when 
y  =  x,  is  the  equivalent  of  Lect.  VI,  §  22. 

f  A  similar  result  for  the  case  of  V(va.  +  a2)  or  V(x2  +  a2). 

£  The  case  of  z=V(a2-y2)  or  V(a2-x*).  Since  T,  R  are  to  be  taken 
on  opposite  sides  of  FG. 


LECTURE    VIII  97 

1 6.  Let  AFB  be  any  curve,  of  which  AD  is  the  axis  and 
DB  is  applied  to  AD;   also  let    VGC  be  another  curve  so 
related  that,  if  any  straight  line  ZF  is  drawn  through  some 
fixed  point  Z  in  the  axis  AD,  and  through  F  a  straight  line 
EFG  is  drawn  parallel  to  DBC,  EG  is  equal  to  ZF;  also  let 
FQ  be  at  right  angles  to  the  curve  AFB;  along  AD,  in  the 
direction  ZE,  take  QR  =  ZE;  then  RG,  being  joined,  will  be 
perpendicular  to  the  curve  VGC. 

[For  this,  Barrow  makes  use  of  the  hyperbola  of  Lect.  VI, 
§  25,  as  the  auxiliary  curve;  he  did  not  give  a  proof  of  the 
theorem  of  that  article,  but  left  it  "  to  the  reader."] 

17.  Let  DP  be  a  straight  line,  and  DRS,  DYX  two  curves 
so  related  that,  if  any  straight  line  REY  is  drawn  parallel  to 
a  straight  line  DB,  given  in  position,  cutting  DP  in  E  and 
the  curves  DRS,  DYX  in  R,  Y,  the  ratio  RY:  DY  is  always 
equal  to  the  ratio   DY:EY;  also  let  the  straight  line   RF 
touch  the  curve   DRS  at  R.     It  is  required  to  draw   the 
tangent  to  the  curve  DYX  at  Y. 

Suppose  the  line  DYO  is  such  that,  if  any  straight  line 
GO  is  drawn  parallel  to  DB,  cutting  the  lines  FR,  FP,  DYO 
in  the  points  G,  P,  0,  and  DO  is  joined,  GO  :  DO  =  DO  :  PO ; 
then  the  curve  DYO  touches  the  curve  DYX  at  Y. 

But,  in  Lect.  VI,  §  12,  it  has  already  been  shown  that 
the  curve  DYO  is  a  hyperbola ;  let  YS  touch  the  hyperbola ; 
then  YS  also  touches  the  curve  DYX. 

NOTE.  If  the  curve  DRS  is  a  circle,  and  the  angle  GDB 
is  a  right  angle,  the  curve  DYX  is  the  ordinary  Cissoid ;  and 
thus  the  tangent  to  it  (together  with  many  other  curves 
similarly  produced)  is  determined. 

7 


98  BARROW'S  GEOMETRICAL  LECTURES 

1 8.  Let  DB,  VK  be  two  lines  given  in  position,  and  let 
the  curve  DYX  be  such  that,  if  from  the  point  D  any  straight 
line  DYH  is  drawn,  cutting  the  straight  line  BK  in  H  and  the 
curve  DYX  in  Y,  the  chord  DY  is  always  equal  to  the  straight 
line  BH  ;  it  is  required  to  draw  the  straight  line  touching 
the  curve  DYX  in  Y. 

With  centre  D  and  radius  DB,  describe  the  circle  BRS; 
let  YER,  drawn  parallel  to  KB,  meet  the  circle  in  R;  join 
DR.  Then  RY :  YD  =  YD:DE;  hence,  the  straight  line 
touching  the  curve  DYX  can  be  found  by  the  preceding 
proposition.* 

19.  Let  DB,  BK  be  two  straight  lines  given  in  position; 
also  let  BXX  be  a  curve  such  that,  if  from  a  point  D  any 
straight  line  is  drawn,  cutting  BK  in  H  and  the  curve  BXX 
in  X,  HX  is  always  equal  to  BH  ;  it  is  required  to  draw  the 
tangent  to  the  curve  BXX  at  X. 

Suppose  that  DYY  is  a  curve  such  that  DY'is  always  equal 
to  BH  (such  as  we  considered  in  the  previous  proposition), 
and  let  YT  touch  this  curve  in  Y,  and  cut  BK  in  R ;  then  let 
the  hyperbola  NXN  be  described,  with  asymptotes  RB,  RT, 
to  pass  through  X;  then  the  hyperbola  NXN  touches  the 
curve  BXX  at  X.  Thus,  if  the  tangent  to  the  hyperbola, 
XS  is  drawn ;  XS  will  also  touch  the  curve  BXX. 

However,  we  seem  to  have  trifled  with  this  succession  of 
theorems  quite  long  enough  for  one  time ;  we  will  leave  off 
for  a  while. 


*  In  §  17,  it  is  not  essential  that  the  curve  RS  should  pass  through  D  ; 
hence  this  statement  is  justifiable. 


LECTURE    VIII  99 


NOTE 

In  the  footnote  to  §  9, 1  state  that  it  is  in  this  theorem  that 
Barrow  starts  his  real  work  on  the  infinitesimal  calculus. 
Certainly  he  has  given  theorems  on  tangents  before  this 
point,  which  have  had  analytical  equivalents;  but  these 
have  been  special  cases.  Here  for  the  first  time  he  gives 
theorems  that  are  equivalent  to  the  differentiation  of  general 
functions,  not  only  of  the  variable  simply,  but  of  any  other 
function  that  is  itself  a  function  of  the  variable.  Thus,  the 
theorem  of  Lect.  VI,  §  22  is  indeed  equivalent  to  the  differ- 
entiation of  ,J(x2-a2)  with  regard  to  x\  but  it  is  in  the 
theorem  of  Lect.  VIII,  §  13  that  he  gives  the  equivalent 
to  the  differentiation  of  *J(y*  -  a2)  with  regard  to  x,  where 
y  is  any  function  of  x  whose  gradient  is  known.  Thus 
Barrow  substantiates  the  last  words  of  the  paragraph  with 
which  he  opens  the  lecture :  "Now  let  us  get  to  it." 

He  however  omits  a  theorem,  which  would  seem  to  fall 
naturally  into  place  in  this  lecture,  as  a  generalization  of 
the  theorem  of  §  1 1 . 

If  XEM,  YFN,  ZGO  are  three  curves  and  PD  is  any  straight 
line  such  that,  if  any  straight  line  PEFG  is  drawn  parallel 
to  a  straight  line  given  in  position,  the  intercept  PE  is 
always  equal  to  FG  ;  also  let  El,  FK  touch  two  of  the  curves 
XEM,  YFN  ;  draw  the  straight  line  GL  such  that,  if  any 
straight  line  HO  is  drawn  parallel  to  DEFG,  cutting  the 
given  lines  as  shown,  KL  =  HI;  then  LG  will  touch  the 
curve  LGO. 

For,  if  the  two  curves  XEM,  YFN  are 
both  convex  to  the  line  VP, 
since          HM  -  NO,   and   HI  -  KL, 
KO>NO>HM>HI>KL; 
hence  the  curve  lies  altogether  above  the 
line  GL 

If  both  curves  are  concave  to  VP,  the 
argument  is  similar,  but  now  ZGO  falls 
altogether  below  the  line  GL 

If  one  of  the  curves  is  concave  and  v 
the  other  convex  to  VP,  say  XEM,  jyF>z,  draw  the  curve 
YFN  so  that  the  intercept  KN  is  always  equal  to  the  inter- 
cept «K ;  then  the  two  curves  YFN,j'F«  touch  and  have  a 


ioo  BARROW'S  GEOMETRICAL  LECTURES 

common  tangent  KF.  Let  now  the  third  curve  be  zGo ; 
then,  since  LO  =  IM  +  KM,  and  L?  =  IM  -  KN,  therefore  00 
is  always  equal  to  2KN;  hence,  by  §  3  above,  the  curves 
zG0,  ZGO  also  touch,  and  LG  is  the  common  tangent. 
Therefore  the  construction  holds  in  this  case  also. 

I  believe  the  omission  of  the  theorem  was  intentional; 
and  I  argue  from  it  that  Barrow  himself  was  not  completely 
satisfied  with  the  theorems  of  §§  3,  4,  thus  corroborating 
my  footnote.  This  theorem  is  of  course  equivalent  to  the 
differential  of  a  sum.  Barrow  may  have  thought  it  evident, 
or  he  may  have  considered  it  to  be  an  immediate  con- 
sequence of  his  differential  triangle ;  but  I  prefer  to  think 
that  he  considered  it  as  a  corollary  of  the  theorem  of  §  5. 
For  this  may  be  given  analytically  as  : — 

If  nw  =  ry  +  (n  —  r)z,  then  n .  dwjdx  =  r .  dyjdx  +  (n  —  r). 
dzjdx.  If  we  take  one  curve  a  straight  line,  and  this  straight 
line  as  the  axis,  we  have  d(ky)jdx  =  k .  dy/dx,  or  the  sub- 
tangents  of  all  "multiple"  curves  have  the  same  subtangent 
as  the  original  curve.  Hence  the  constructions  for  the 
tangents  to  "sum"  and  "difference"  curves  follow 
immediately : — 

Let  A  A  A,  BBB  be  any  two  curves •,  of  which  EF  is  taken 
as  a  common  axis;  let  NAB  be  any  straight  line  applied 
perpendicular  to  EF;  let  the  tangents  AS,  BR,  cut  EF  in 
S,  R;  take  ha,  B£  equal  to  NA,  NB  respectively,  and  also  let 
NC=  NA  +  NB,  <w«*ND  =  NA-NB. 

Join  Sa,  R<£  intersecting  in  T,  and  draw  TV  perpendicular 
/0EF. 

Then  TC  will  touch  the  "sum"  curve  CCC,  and  VD  will 
touch  the  "  difference  "  curve  ODD. 

It  seems  rather  strange,  considering  Barrow's  usual  custom, 
that  he  fails  to  point  out  that,  in  §  12,  if  the  curve  XEM  is  a 
circle  passing  through  D,  the  curve  YFN  is  the  Cardioid  or 
one  of  the  other  LimaQOns. 

The  final  words  of  the  lecture  seem  to  indicate  that 
Barrow  now  intends  to  proceed  to  what  he  considers  to 
be  the  really  important  part  of  his  work ;  and,  in  truth, 
this  is  what  the  next  lecture  will  be  found  to  be. 


LECTURE    IX 

Tangents  to  curves  formed  by  arithmetical  and  geometrical 
means.  Paraboliforms.  Curves  of  hyperbolic  and  elliptic 
form.  Differentiation  of  a  fractional  power ;  products  and 
quotients. 

We  will  now  proceed  along  the  path  upon  which  we 
started. 

i.  Let  the  straight  lines  AB,  VD  be  parallel  to  one  another ; 
and  let  DB  cut  them  in  a  given  position ;  also  let  the  lines 
EBE,  FBF  pass  through  B,  being  so  related  that,  if  any 
straight  line  PG  is  drawn  parallel  to  DB,  PF  is  always  an 
arithmetical  mean  of  the  same  order  between  PG  and  PE ; 


Fig.  94. 

and  let  the  straight  line  B8  touch  the  curve.     Required  to 
draw  the  tangent  at  B  to  the  curve  FBF. 


102  BARROW'S  GEOMETRICAL  LECTURES 

Let  tli-5  numbers  N,  M  (as  explained  in  Lect.  VII,  §  12) 
be  the  exponents  of  the  proportionals  PF,  PE ;  take  DT, 
such  that  N  :  M  =  D8 :  DT,  and  join  TB ;  then  TB  touches 
the  line  FBF. 


For,  in  whatever  position  the  line  PG  is  drawn,  cutting 
the  given  lines  as  shown  in  the  figure,  we  have 
FG  :  EG  =  N  :  M  =  DS  :  DT  =  LG  :  KG. 

Hence,  since  by  hypothesis  KG  <  EG,  .'.  LG  <  KG  ;  and 
thus  it  has  been  shown  that  the  straight  line  TB  falls  wholly 
without  the  curve  FBF.* 

2.  All  other  things  remaining  the  same,  let  now  PF  be 
a  geometrical  mean  between  PG  and  PE  (namely,  the  mean 
of  the  same  order  as  in  the  former  case  of  the  arithmetical 
mean)  ;  then  the  same  straight  line  touches  the  curve  FBF. 

For  the  lines  constructed  in  this  way  from  arithmetical 
and  geometrical  means  touch  one  another  ;  hence,  since  BT 
touches  the  one  curve,  it  will  also  touch  the  other.  f 

Example.  —  Suppose  PF  is  the  third  of  six  means  between 
PG  and  PE,  then  M  =  7,  and  N  =  3  ;  and  DS  :  DT  =  3  :  7. 

*  Note  that  in  this  case,  FG  :  EG  =  LG  :  KG  ;  and  thus  this  is  a  par- 
ticular case  of  the  curves  in  Lect.  VIII,  §  5;  the  analytical  equivalent  is 


f  Analytical  equivalent  :  —  If  y  is  any  function  of  x,  and  zn  =  an-r  .yr,  then 
dz\dx  =  (r\ri)  .  dyjdx,  when  z  =  y  =  a. 


LECTURE  IX 


103 


3.  Again,  the  preceding   hypothesis   being  made   in  all 
other  respects,  let  any  point  F  be  taken  in  the  curve  FBF; 

Y 

X 

V 


Fig.  95- 

then  a  straight  line  touching  the  curve  may  be  drawn  by  a 
similar  method.  Thus,  let  the  straight  line  PG  be  drawn 
through  F  parallel  to  DB,  cutting  the  curve  EBE  in  E, 
and  let  EX  touch  the  curve  EBE  at  E ;  take  PY,  such  that 
N  :  M  =  PX  :  PY,  and  join  FY. 
Then  the  straight  line  FY  touches  the  curve  FBF.* 
For,  if  through  E  the  straight  line  CEI  is  drawn  parallel 
to  AB  or  YD,  and  it  is  supposed  that  a  curve  HEH  passing 
through  E  is  such  that,  if  any  straight  line  Ql  is  drawn 
parallel  to  DB,  cutting  the  curves  EBE,  HEH  in  L,  H,  and 
the  straight  lines  CE,  VP  in  I,  Q,  QH  is  a  mean  between 
Ql  and  QL  of  the  same  order  as  PF  was  between  PG  and 
PE ;  then  it  follows  from  the  preceding  proposition  that, 
if  YE  is  joined,  it  will  touch  the  curve  HEH. 

But  the  curves  HEH,  FBF  are  analogous  curves  (Lect.  VII, 
§  7);  therefore  YF  touches  the  curve  FBF  (Lect.  VIII,  §  5). 

*  This  is  a  generalization  of  the  last  theorem  ;  the  equivalent  is  that, 
in  general,  if  zn  =  a1*-* ,yr,  then  (i/z).  dzjdx  =  (n/r) .  (\\y}.dy\dx.  The 
analogy  of  the  curves  occurs  in  the  case  of  the  arithmetical  means,  for  then 
IH  :  HL  =  GF  :  FE. 


104  BARROWS  GEOMETRICAL  LECTURES 

4.  Note  that,  if  the  line  EBE  is  supposed  to  be  straight, 
then  the  line  FBF  is  one  of  the  parabolas  or  curves  of  the 
form  of  a  parabola  ("  paraboliforms  ").    Therefore,  that  which 
is   usually  held   to   be  "known"  concerning  these  curves 
(deduced  by  calculation  and  verified  by  a  sort  of  induction, 
but  not,  as  far  as  I  am  aware,  proved  geometrically)  flows 
from  an  immensely  more  fruitful  source,  one  which  covers 
innumerable  curves  of  other  kinds.* 

5.  Hence  the  following  deductions  are  evident : — 

If  TD  is  a  straight  line  and  EEE,  FFF  are  two  curves  so 
related  that,  when  straight  lines  PEF  are  drawn  parallel  to 
BD,  a  straight  line  given  in  position,  the  ordinates  PE  vary 
as  the  squares  on  the  ordinates  PF;  and  if  E8,  FT,  straight 
lines  drawn  from  the  ends  of  the  same  common  ordinate, 
touch  these  curves ;  then  TP  =  28P.  But,  if  the  ordinate 
PE  varies  as  the  cube  of  PF,  then  TP  =  38P;  if  PE  varies 
as  the  fourth  power  of  PF,  then  TP  =  48  P ;  and  so  on  in 
the  same  manner  to  infinity.! 

6.  Again,  let  AGB  be  a  circle,  with  centre  D  and  radius 
DB,  and  let  EBE,  FBF  be  two  lines  passing  through  B,   so 
related  to  one  another  that,  when  any  straight  line  DG  is 
drawn  through   D,   DF  is  always  an  arithmetical  mean  of 
the  same  order  between  DG  and  DE ;  also  let  the  straight 

*  See  note  at  the  end  of  this  lecture  ;  where  it  is  shown  that  this  theorem 
is  equivalent  to  a  rigorous  demonstration  of  the  method  for  differentiating 
a  fractional  power  of  the  variable. 

f  This  is  a  special  case  of  the  preceding  theorem  ;  for  PF  is  the  simple 
geometrical  mean  between  PE  and  a  definite  length  PG  ;  or  the  second  of 
two,  the  third  of  three,  etc. ,  geometrical  means  between  PE  and  PG  ;  thus 
PF2  =  PE.PG,  PF3  =•  PE,  PG2,  etc.  This  enables  Barrow  to  differentiate 
any  power  or  root  of/(jr),  when  he  can  differentiate  f(x]  itself. 


LECTURE  IX  105 

line  BO  touch  the  curve  EBE  at  B;  required  to  draw  the 
tangent  at  B  to  the  curve  FBF. 

This  (demonstrated  generally,  to  a  certain  extent,*  in 
Lect.  VIII,  §  8)  will  here  be  specially  shown  to  follow  more 
clearly  and  completely  from  the  method  above.  Thus  : —  ,• 

Let  DQ  be  perpendicular  to  DB,  cutting  BO  in  8  ;  and 
let  N  :  M  =  DS  :  DT  ;  join  BT.  Then  BT  touches  the  curve 
FBF.t 

7.  Hence,  other  things  remaining  the  same  as   before, 
if  the  straight  line   DF  is  always  taken  as  a  geometrical 
mean  (of  the  same  order  as  before)  between  DG  and  DE, 
the  same  straight  line  BT  will  touch  the  curve  FBF  also. 

For  the  lines  formed  from  arithmetical  means  and  from 
geometrical  means  of  the  same  order  touch  one  another, 
and  have  a  common  tangent. 

8.  Further,  other  things  remaining  the  same  as  in  the 
preceding  proposition,  let  any  point  P  be  taken  in  the  curve 
FBF;  then  the  straight  line  that  touches  the  curve  at  this 
point  can  be  determined  by  a  similar  plan. 

Let  the  straight  line  DF  be  drawn,  cutting  the  curve  EBE 
in  E ;  also  draw  DQ  perpendicular  to  DG  cutting  EO  the 
tangent  to  EBE  in  X  ;  make  DX  :  DY  =  N  :  M  ;  join  EY,  and 
draw  FZ  parallel  to  EY.  Then  FZ  touches  the  curve  FBF. 

Hence,  not  only  the  tangents  to  innumerable  spirals, 
but  also  those  to  a  boundless  number  of  others  of  different 
kinds,  can  be  determined  quite  readily. 

*  The  actual  construction  for  the  asymptotes  or  tangent  to  the  auxiliary 
hyperbola  is  not  given. 

t  Barrow  proves  his  construction  by  the  use  of  an  auxiliary  hyperbola 
using  Lect.  VI,  §  4,  and  VIII,  §  9. 


io6  BARROW'S  GEOMETRICAL  LECTURES 


9.  Hence,  it  is  clear  that,  if  two  lines  EEE,  FFF  are  so 
related  that,  when  any  straight  line  DEF  is  drawn  from  a 
fixed  point  D,  DE  varies  as  the  square  on  DF ;  and  if  ES, 
FT  are  the  tangents  to  the  curves  at  the  ends  E,  F,  meeting 
the  line  perpendicular  to   DEF  in    S,  T;   then  DT  -  2DS. 
But,  if  DE  varies  as  the  cube  of  DF,  DT  =  3DS ;  and  so  on.* 

10.  Let  YD,  TB  be  two  straight  lines  meeting  in  T,  and 
let  a  straight  line  BD,  given  in  position,  fall  across  them ; 

T 
V 
S 


Fig.   100. 

also  let  the  lines  EBE,  FBF  pass  through  B  and  be  such 
that,  if  any  straight  line  PG  is  drawn  parallel  to  BD,  PF  is 
always  an  arithmetical  mean  of  the  same  order  between  PG 
and  PE;  also  let  BR  touch  the  curve  EBE.  Required  to 
draw  the  tangent  at  B  to  the  curve  FBF. 

Taking  numbers  N,  M  to  represent  the  exponents  of  PF, 
PE,  make  N  .  TD  +  (M  -  N) .  RD  :  M  .  TD  =  RD  :  SD,  and  join 
BS;  then  B8  touches  the  curve  FBF.f 

*  As  the  theorems  of  §§  6,  7,  8,  9  are  only  the  polar  equivalents  of  §§  i, 
2,  3,  5,  the  figures,  and  proofs  are  not  given  ;  their  inclusion  by  Barrow 
suggests  that  he  was  aware  of  the  fact  that,  with  the  usual  modern  notation, 
tan  0  =  r.  dtydr. 

•j*  The  form  of  the  equation  suggests  logarithmic  differentiation  :  see  note 
at  end  of  this  lecture. 


LECTURE  IX  107 

For,,  if  any  straight  line  PG  is  drawn,  cutting  the  given 
lines  as  in  the  figure,  we  have  EG  :  FG  =  M  :  N  ; 
therefore         FG  .  TD  :  EG  .  TD •  =  N  .  TD  :  M  .  TD ; 
also  EF  .  RD  :  EG  .  TD  =  (M  -  N) .  RD  :  M  .  TD. 

Hence,  adding  the  antecedents,  we  have 
FG.TD  +  EF.RD:EG.TD  =  N.TD  +  (M  -  N).RD:M.TD 

=  RD:SD. 

Now,  LG  .  TD  +  EF .  RD  :  EG  . TD  =  RD  :  SD;  VII,  §  4, 
therefore 

FG.TD+EF.RD:EG.TD  =  LG  .TD  +  KL.  RD :  KG  .TD 

Hence,  since  EG  >  KG, 

.-.FG.TD  +  EF.RD>  LG  .  TD  +  KL.  RD* 
.-.  ratio  compounded  of  FG/EF  and  TD/RD  >  than 

that  compounded  of  LG/KL  and  TD/RD 
or,  removing  the  common  ratio  RD/TD, .'.  EG/EF  >  LG/KL; 
hence,  by  componendo  EG/EF  >  KG/KL  >  EG/EL  (by  Lect. 
VII,  §  i);  therefore  EF  <  EL,  or  the  point  L  is  situated 
on  the  far  side  of  the  curve  FBF;  and  thus  the  problem 
is  solved. 

n.  Moreover,  all  other  things  remaining  the  same,  if 
PF  is  supposed  to  be  a  geometrical  mean  of  the  same  order 
(plainly  as  in  the  cases  just  preceding)  the  same  straight  line 
BS  will  touch  the  curve  FBF. 


*  This  is  either  a  very  bad  slip  on  Barrow's  pKrt,  or  else  he  is  making  the 
unjustifiable  assumption  that  near  B  the  ratio  of  LK  to  FE  is  one  of  equality. 
In  either  case  the  proof  cannot  be  accepted.  The  demonstration  can,  how- 
ever, be  completed  rigorously  as  follows  from  the  line 

FG .  TD  +  EF .  RD  :  EG  .  TD  =  LG .  TD  +  KL .  RD  :  KG .  TD. 
Hence  EG/EF  :  KG/KL=  FG/EF  +  RD/TD  :  LG/KL  +  RD/TD 

=  EF/EF  -  RD/TD  :  KL/KL  -  RD/TD  (dividendo)  \ 

therefore  EG/EF  =  KG/KL,     or     EG/FG  =  KG/GL  ;  hence,  since  EG  >  KG. 
it  follows  that  FG  >  LG,  i.e.  L  falls  without  the  curve. 


io8  BARROWS  GEOMETRICAL  LECTURES 

Eocamplc. — If  PF  is  a  third  of  six  means,  or  M  =  7,  N  =  3  ; 
then 
3TD  +  4RD:7TD  =  RD:8D,    i.e.    SD  =  7TD.RD/(3TD  +  4RD). 

12.  It  is  evident  that,  if  any  point  F  whatever  is  taken  on 
the  line  FBF,  the  tangent  at  F  can  be  drawn  in  a  similar 
manner.     Thus,  through  F  draw  the  straight  line  PG  par- 
allel to  DB,  cutting  the  curve  EBE  at  E,  and  through  E  let 
ER  be  drawn  touching  the  curve  EBE  at  E ;  then  make 

N.TP  +  (M-N).RP:M.TP  =  RP:SP, 
and  join  SF.     Then  8F  will  touch  the  curve  FBF. 

13.  Note  that,  if  EBE  is  a  straight  line  (i.e.  coinciding 
with  the  straight  line  BR),  the  line  FBF  is  one  of  an  infinite 
number  of  hyperbolas  or  curves  of  hyperbolic  form ;  and 
we  have  therefore  included  in  the  one  theorem  a  method  of 
drawing  tangents  to  these,  together  with  innumerable  others 
of  different  kinds. 

14.  If,  however,  the  points  T,  R  do  not  fall  on  the  same 
side  of  D  (or  P),  the  tangent  BS  to  the  curve  EBF  is  drawn 
by  making  N.RD-(M-N).TD:M.TD  =  RD  :  SD. 

15.  Hence  also  the  tangents  to  not  only  all  elliptic  curves 
(in  the  case  when  EBE  is  supposed  to  be  a  straight  line 
coinciding  with   BR),   but   to   an   innumerable  number  of 
other  curves  of  different  kinds,  can  be  determined  by  the 
one  method. 

Example. — If  PF  is  the  fourth  of  four  means,  i.e.  M  =  5, 
and  N  =  4;  then  SD  =  5TD.  RD/(4RD-TD). 

NOTE.     If  it  happens  that  N  .  RD  =  (M  -  N).TD,  then  DS 


LECTURE  IX  109 

is  infinite  ;  or  BS  is  parallel  to  VD.     Other  points  may  be 
noticed,  but  I  leave  them. 

1 6.  Amongst  innumerable  other  curves,  the  Cissoid  and 
the  whole  class  of  cissoidal  curves  may  be  grouped  together 
byihis  method.  For,  let  DSB  be  a  semi-right  angle;  and 
let  SGB,  SEE  be  two  curves  so  related  that,  if  any  straight 
line  GE  is  drawn  parallel  to  BD,  cutting  the  given  lines 
BS,DS  in  F,  P,  PG,  PF,  PE  are  in  continued  proportion; 
also  let  the  straight  line  GT  touch  the  curve  8G  B  at  G  ;  then 
the  line  touching  the  curve  SEE  is  found  by  making 

2TP-SP:TP  »  SP:RP; 
and,  in  every  case,  if  RE  is  joined,  RE  touches  SEE. 

The  proof  is  easy  from  what  has  gone  before. 

Now,  if  the  curve  SGB  is  a  circle,  and  the  angle  of  appli- 
cation, SPG,  is  a  right  angle,  then  the  curve  SEE  is  the 
ordinary  Cissoid  or  the  Cissoid  of  Diocles;  otherwise  it 
will  be  a  cissoidal  curve  of  some  other  kind.  But  I  "only 
mention  this  in  passing,  and  will  not  now  detain  you  longer 
over  it. 

NOTE 

This  lecture  is  remarkable  for  the  important  note  of  §  4. 
In  it,  Barrow  calls  his  readers'  attention  to  the  fact  that  he 
has  given  a  method  for  drawing  tangents  to  any  of  the 
parabolas  or  paraboliforms ;  and  apparently  he  refers  in 
more  or  less  depreciative  words  to  the  work  of  Wallis, 
whilst  claiming  that  his  own  work  is  a  geometrical  demon- 
stration, and  therefore  rigorous.  If  we  take  a  line  parallel 
to  PG,  and  DV,  as  the  coordinate  axes,  and  suppose  them 
rectangular  or  oblique,  then  PFM  =  PGM~N.PEN  gives 
XM  =  aM~N  ,j>N,  or  y  =  k .  XM/N,  as  the  general  equation  to 
the  curve  FBF. 


i io  BARROW'S  GEOMETRICAL  LECTURES 

Also,  dy\dx  =  PT/PF  =  (PT/PS) .  (PS/PF)  -  (M/N) . y/x; 
or,  if  the  axes  are  interchanged,  the  equation  to  the  curve 
is  y  =  *.*N/Mf  and  then  dy\dx  =  PF/PT  -  (N/M) i.^/a;. 
Note  particularly  that  the  form  suggests  logarithmic  dif- 
ferentiation. 

The  theorem  of  §  6  is  a  particular  case  of  this,  in  which 
N  =  i,  i.e.  PF  is  the  first  of  any  number  of  means  between 
PG  and  PE,  and  the  equations  of  the  curves  arejy  =  &.x2, 
&.x3,  6.x*,  etc.  (the  "parabolas"  as  distinguished  from 
the  "  paraboliforms  "). 

It  seems  strange,  unless  perhaps  it  is  to  be  ascribed  to 
Barrow's  dislike  for  even  positive  integral  indices,  that  he 
does  not  make  a  second  note  to  the  effect  that  if  the  curve 
EBE  is  a  hyperbola  whose  asymptotes  are  VD  and  a  line 
parallel  to  PG,  then  the  curves  FBF  are  the  hyperboliforms. 
For,  from  this  particular  case,  in  a  manner  similar  to  the 
foregoing,  it  follows  that  if  y  =  c .  x~r,  where  r  is  any 
positive  rational,  either  greater  or  less  than  unity,  then 
dyjdx  =  -  r(yjx).  But  Barrow  probably  intends  the  recip- 
rocal theorem  of  Lect.  VIII,  §  9,  to  be  used  thus : — If 
y  =  c.x~r,  let  z  =  ijy  =  k.xr ;  then  from  Lect.  VIII,  §  9, 
we  have  (1/0) .  dz\dx  =  -  (i/y) .  dyjdx ;  also  from  the  above, 
dzfdx=  r.zjx;  hence  dyjdx  =  (  -  r) .  yjx.  I  suggest  that 
Barrow  found  out.  these  constructions  by  analysis,  using 
letters  such  as  a  and  e  instead  of  dy  and  dx,  and  that  the 
form  of  the  results  suggests  very  strongly  that  he  first 
expressed  his  equation  logarithmically. 

Anyway,  Barrow  was  the  first  to  give  a  rigorous  demon- 
stration of  the  form  of  the  differential  coefficient  of  xr, 
where  r  is  any  rational  whatever.  As  far  as  I  am  aware, 
it  is  the  only  proof  that  has  ever  been  given,  that  does  not 
involve  the  consideration  of  convergence  of  infinite  series, 
or  of  limiting  values,  in  some  form  or  other.  Moreover, 
he  gives  it  in  a  form,  which  yields,  as  a  converse  theorem, 
the  solution  of  the  differential  equation  dyjdx  =  r .  dzjdx, 
although,  of  course,  this  is  not  noted  by  Barrow,  simply 
because  he  had  not  the  notation. 

Again,  considering  §  8,  which  is  only  §  3  with  the 
constant  distance  between  the  parallels,  PG,  replaced  by 
the  constant  radius,  DG,  we  see  that,  if  DB  is  the  initial 
line,  and  the  angle  BDG  is  0,  and  the  angles  between  the 


LECTURE  IX  in 

vector  DG  and  the  tangents  at  E  and  F  are  </>  and  x> 
DF  =  R,  DE  ••=  r,  and  DG  =  a,  then  tan  $  :  tan  x  =  N  :  M,  and 
RM  =  aM-N.rN;  hence  (d6/dr)l(d6/dR)  =  dR/dr  =  (1\IM).(Rlr) 
and  ta/z  <£ :  /a«  x  =  r  dOjdr :  R  .  ddjdR  ;  and  I  suggest  that  'it 
was  thus  that  Barrow  obtained  the  construction  for  this 
theorem.  I  go  further.  Although  it  is  a  consequence  of 
a  consideration  of  the  whole  work,  the  present  place  is  the 
most  convenient  one  for  me  to  state  my  firm  conviction 
that  Barrow's  drawing  of  tangents  was  a  result  of  his 
knowledge  of  the  fundamental  principles  of  a  calculus 
of  infinitesimals  in  an  algebraic  form,  which  may  have 
been  so  cumbrous  that  it  was  only  intelligible  to  himself 
when  expressed  in  geometrical  form.  I  fail  to  see  how  else 
he  could  possibly  have  arrived  at  some  of  his  constructions, 
or  elaborated  so  many  of  them  in  the  comparatively  short 
time  that  he  had  to  spend  upon  them ;  unless  indeed  he 
was  a  far  greater  genius  than  even  I  am  trying  to  make 
him  out  to  be.  If  he  had  stumbled  on  the  idea  in  his 
young  days,  as  might  be  possible,  one  could  better  under- 
stand these  theorems  as  being  gradually  evolved ;  but  we 
have  his  own  words  against  this :  "  The  lectures  were 
elicited  by  my  office."  Thus  I  suggest  that  whilst  his 
geometrical  theorems  perhaps  took  definite  shape  whilst 
he  was  Professor  of  Geometry  at  Gresham,  his  knowledge 
of  the  elements  of  the  calculus  dated  from  before  this, 
time. 

Last,  but  by  no  means  least,  the  theorems  of  §§  10,  n, 
12  are  modifications  of  §§  i,  2,  3,  in  which  a  pair  of 
inclined  lines  are  substituted  for  the  pair  of  parallels. 
Referring  to  Fig.  100  on  p.  106,  take  the  angle  BDT  a 
right  angle,  and  DT  as  the  axis  of  x,  then  the  relation 
given  is  a  relation  between  subtangents  solely.  Further, 
instead  of  BT  we  can  take  a  fixed  curve  touching  BT  at 
B  ;  and  we  have  : — 

If  PF1*  =  PGM- N.PEN,  thenN/RD  +  (M-N)/TD=  M/8D. 

Also,  if  we  take  Z^-i.  PH  =  PFM,  we  have  by  §  5,  if  WD 
is  the  subtangent  to  the  locus  of  H,  1/WD  =  M/SD. 

This  affords  a  complete  rule  for  products,  and  combining 
the  result  with  the  reciprocal  theorem  of  Lect.  VIII,  9, 
for  quotients  also. 


H2  BARROW'S  GEOMETRICAL  LECTURES 

Thus,  putting  N  =  i,  and  M  =  2,  we  have  for  the  general 
theorem  of  §  1 2  the  remarkably  simple  results  : — 

j^GGG,  EEE  are  tivo  curves,  and  PEG  is  a  straight  line 
applied  perpendicular  to  an  axis  PRT,  and  GT,  ER  are  the 
tangents  to  GGG  and  EEE,  then 

(i)  If  HHH  is  another  curve,  so  related  to  the  other  two 
that  Z.  PH  =  PE.  PG  ;  then,  if  HW  is  the  tangent  to  HHH, 
meeting  the  axis  in  W,  1/PW  =  1/PR+l/PT;  i.e.  PW  is  a 
fourth  proportional  to  PR  +  PT,  PR,  and  PT. 

(it)  If  KKK  is  another  curve  so  related  to  GGG  and  EEE 
that  PK:Z  =  PE:  PG,  then,  if  KV  is  the  tangent  to  KKK, 
meeting  the  axis  in  V,  1/PV  =  1/PR-l/PT;  or  PV  is  a 
fourth  proportional  to  PT-  PR,  PR,  and  PT. 

The  elegance  of  the  geometrical  results  probably  accounts 
for  the  fact  that  Barrow  adheres  to  the  subtangent,  as  used 
by  Descartes,  Fermat,  and  others ;  and  this  would  tend  to 
keep  from  him  the  further  discoveries  and  development  that 
awaited  the  man  who  considered,  instead  of  the  subtangent, 
the  much  more  fertile  idea  of  the  gradient,  as  represented 
by  Leibniz'  later  development,  dy\dx ;  the  germ  of  the  idea 
of  the  gradient  is  of  course  contained  in  the  "a  and  e" 
method,  but  it  is  neglected. 

Note  the  disappearance  of  the  constant  Z;  hence  the 
curves  may  be  drawn  to  any  convenient  scale,  which  need 
not  be  the  same,  for  all  or  any,  in  the  direction  parallel  to 
PEG.  The  analytical  equivalents  are : — 

(i)  If  w  =  yz,  then  (\\w)dw\dx  =  (i/y)dy/dx-{-(i/z)dz/dx ; 
(ii)  if  v  =  y/z,  then  (i/v)dv/dx  =  (ijy)dy/dx-  (i/z)dz/dx. 

The  first  of  these  results  is  generally  given  in  modern  text- 
books on  the  calculus,  but  I  do  not  remember  seeing  the 
second  in  any  book.  Thus,  for  products  and  quotients  we 
may  state  the  one  rule  : — 

if  ,_  uv    dy  —  u<v[l  du     i      dv  _  i      dw  _  i  dz~\ 
wz  dx     wz\u  dx     v     dx     w  '  doo     z  dx\ 
where  u,  v,  w,  z,  and  y  are  all  functions  of  x. 


LECTURE  X 

Rigorous  determination  of  dsjdx.  Differentiation  as  the 
inverse  of  integration.  Explanation  of  the  "  Differential 
Triangle"  method;  with  examples.  Differentiation  of  a 
trigonometrical  function. 

1.  Let  AEG   be   any  curve   whatever,   and   API    another 
curve  so  related  to  it  that,  if  any  straight  line  EF  is  drawn 
parallel  to  a  straight  line  given  in  position  (which  cuts  AEG 
in  E  and  API  in  F),  EF  is  always  equal  to  the  arc  AE  of  the 
curve  AEG,  measured  from  A;  also  let  the  straight  line  ET 
touch  the  curve  AEG  at   E,  and   let   ET  be  equal  to  the 
arc  AE;  join  TF;  then  TF  touches  the  curve  AFI. 

2.  Moreover,  if  the  straight  line  EF  always  bears  any  the 
same  ratio  to  the  arc  AE,  in  just  the  same  way  FT  can  be 
shown  to  touch  the  curve  AFI.* 

3.  Let  AGE  be  any  curve,  D  a  fixed  point,  and  AIF  be 
another  curve  such  that,  if  any  straight  line  DEF  is  drawn 
through  D,  the  intercept  EF  is  always  equal  to  the  arc  AE; 
and  let  the  straight  line  ET  touch  the  curve  AGE;  make 

*  Since  the  arc  is  a  function  of  the  ordinate,  this  is  a  special  case  of 
the  differentiation  of  a  sum,  Lect.  IX,  §  12  ;  it  is  equivalent  lo  d(as+y)/dx  = 
a  .  dsjdx  +  dyldx  ;  see  note  to  §  5. 

8 


ii4  BARROWS  GEOMETRICAL  LECTURES 

TE  equal  to  the  arc  AE  *  ',  let  TKF  be  a  curve  such  that,  if 
any  straight  line  DHK  is  drawn  through  D,  cutting  the  curve 
TKF  in  K  and  the  straight  line  TE  in  H,  HK  =  HT;  then  let 
FS  be  drawn  f  to  touch  TKF  at  F;  F8  will  touch  the  curve 
AlFalso. 

4.  Moreover,  if  the  straight  line  EF  is  given  to  bear  any 
the  same  ratio  to  the  arc  AE,  the  tangent  to  it  can  easily  be 
found  from  the  above  and  Lect.  VIII,  §  8. 

5.  Let  a  straight  line  AP  and  two  curves  AEG,  AFI  be  so 
related  that,  if  any  straight  line  DEF  is  drawn  (parallel  to 

*'     HG 


B 

T 
Fig.  1 06. 

AB,  a  straight  line  given  in  position),  cutting  AP,  AEG,  AFI, 
in  the  points  D,  E,  F  respectively,  DF  is  always  equal  to  the 
arc  AE;  also  let  ET  touch  the  curve  AEG  at  E;  take  TE 
equal  to  the  arc  AE,  and  draw  TR  parallel  to  AB  to  cut 
AP  in  R ;  then,  if  RF  is  joined,  RF  touches  the  curve  AFI. 

For,  assume  that  LFL  is  a  curve  such  that,  if  any  straight 
line  PL  is  drawn  parallel  to  AB,  cutting  AEG  in  G,  TE  in  H, 
and  LFL  in  L,  the  straight  line  PL  is  always  equal  to  TH 
and  HG  taken  together.  Then  PL  >  arc  AEG  >  PI ;  and 

*  TE,  AE  are  drawn  in  the  same  sense, 
f  By  Lect.  VIII,  §  19. 


LECTURE  X  115 

therefore  the  curve  LFL  touches  the  curve  API.  Again, 
by  Lect  VI,  §  26,  PK  =  TH  (or  KL  =  GH) ;  hence  the  curve 
LFL  touches  the  line  RFK  (by  Lect.  VII,  §  3);  therefore 
the  line  RFK  touches  the  curve  AFI.* 

6.  Also,  if  DF  always  bears  any  the  same  ratio  to  the 
arc  AE,  RF  will  still  touch  the  curve  AFI ;  as  is  easily  shown 
from  the  above  and  Lect.  VIII,  §  6. 

7.  Let  a  point  D  and  two  curves  AGE,  DFI  be  so  related 
that,  if  any  straight  line  DFE  is  drawn  through  D,  the  straight 
line  DF  is  always  equal  to  the  arc  AE ;  also  let  the  straight 
line  ET  touch  the  curve  AGE  at  E;  make  ET  equal  to  the 
arc  AE;  and  assume  that  DKK  is  a  curve  such  that,  if  any 
straight  line  DH  is  drawn  through  D,  cutting  DKK  in  K  and 
TE  in  H,  the  straight  line  DK  is  always  equal  to  TH.     Then, 
if  FS  is  drawn  (by  Lect.  VIII,  §    16)  to  touch  the  curve 
DKK  at  F,  FS  touches  the  curve  DIF  also. 

8.  Moreover,  if  DF  always  bears  any  the  same  ratio  to  the 
arc  AE,  the  straight  line  touching  the  curve  DIF  can  likewise 
be  drawn ;  and  in  every  case  the  tangent  is  parallel  to  FS. 

9.  By  this  method  can  be  drawn  not  only  the  tangent 
to  the  Circular  Spiral,  but  also  the  tangents  to  innumerable 
other  curves  produced  in  a  similar  manner. 

10.  Let  AEH  be  a  given  curve,  AD  any  given  straight  line 


*  The  proof  of  this  theorem  is  given  in  full,  since  not  only  is  it  a  fine 
example  of  Barrow's  method,  but  also  it  is  a  rigorous  demonstration  of  the 
principle  of  fluxions,  that  the  motion  along  the  path  is  the  resultant  of  the 
two  rectilinear  motions  producing  it.  Otherwise,  for  rectangular  axes, 
(dsldxf  =  i  +  (dy/dx)z;  for  ds\dx  =  DF/DR  =  ET/DR  =  Cosec  DET  and 
dy\dx=Cot  DET. 


ii6  BARROW'S  GEOMETRICAL  LECTURES 

in  which  there  is  a  fixed  point  D,  and  DH  a  straight  line  given 
in  position ;  also  let  AGB  be  a  curve  such  that,  if  any  point 
G  is  taken  in  it,  and  through  G  and  D  a  straight  line  is 
drawn  to  cut  the  curve  AEH  in  E,  and  GF  is  drawn  parallel 
to  DH  to  cut  AD  in  F,  the  arc  AE  bears  to  AF  a  given  ratio, 
X  to  Y  say ;  also  let  ET  touch  the  curve  AEH  ;  along  ET  take 
EV  equal  to  the  arc  AE  ;  let  OGO  be  a  curve  such  that,  if  any 
straight  line  DOL  is  drawn,  cutting  the  curve  OGO  in  0  and 
ET  in  L,  and  if  OQ  is  drawn  parallel  to  GF,  meeting  AD  in 
Q,  LV  :  AQ  =  X  :  Y.  Then  the  curve  OGO  is  a  hyperbola  (as 
has  been  shown). *  Then,  if  GS  touches  this  curve,  G8  will 
touch  the  curve  AGB  also. 

If  the  curve  AEH  is  a  quadrant  of  a  circle,  whose  centre 
is  D,  the  curve  AGB  will  be  the  ordinary  Quadratrix.  Hence 
the  tangent  to  this  curve  (together  with  tangents  to  all 
curves  produced  in  a  similar  way)  can  be  drawn  by  this 
method. 

I  meant  to  insert  here  several  instances  of  this  kind ; 
but  really  I  think  these  are  sufficient  to  indicate  the 
method,  by  which,  without  the  labour  of  calculation,  one 
can  find  tangents  to  curves  and  at  the  same  time  prove  the 
constructions.  Nevertheless,  I  add  one  or  two  theorems, 
which  it  will  be  seen  are  of  great  generality,  and  not  lightly 
to  be  passed  over. 

n.  Let  ZGE  be  any  curve  of  which  the  axis  is  AD;  and 
let  ordinates  applied  to  this  axis,  AZ,  PG,  DE,  continually 


*  Only  proved  for  a  special  case  in  Lect.  VI,  §  17  ;  but  the  method  can 
be  generalized  without  difficulty. 


LECTURE  X 


117 


increase  from  the  initial  ordinate  AZ;  also  let  AIF  be  a  line 
such  that,  if  any  straight  line  EOF  is  drawn  perpendicular 
to  AD,  cutting  the  curves  in  the  points  E,  F,  and  AD  in  D,  the 
rectangle  contained  by  DF  and  a  given  length  R  is  equal 
to  the  intercepted  space  ADEZ;  also  let  DE  :  DF  =  R  :  DT, 
and  join  DT.  Then  TF  will  touch  the  curve  AIF. 


Fig.  109. 

For,  if  any  point  I  is  taken  in  the  line  AIF  (first  on  the 
side  of  F  towards  A),  and  if  through  it  IG  is  drawn  parallel 
to  AZ,  and  KL  is  parallel  to  AD,  cutting  the  given  lines  as 
shown  in  the  figure ;  then  LF  :  LK  =  DF  :  DT  =  DE  :  R,  or 
R.LF=  LK.DE. 

But,  from  the  stated  nature  of  the  lines  DF,  PK,  we  have 
R .  LF  =  area  PDEG ;  therefore  LK .  DE  =  area  PDEG  <  DP . DE ; 
hence  LK  <  DP  <  LI. 

Again,  if  the  point  I  is  taken  on  the  other  side  of  F,  and 
the  same  construction  is  made  as  before,  plainly  it  can  be 
easily  shown  that  LK  >  DP  >  LI. 

From  which  it  is  quite  clear  that  the  whole  of  the  line 
TKFK  lies  within  or  below  the  curve  AIFI. 

Other  things  remaining  the  same,  if  the  ordinates,  AZ, 
PG,  DE,  continually  decrease,  the  same  conclusion  is 


n8  BARROW'S  GEOMETRICAL  LECTURES 

attained  by  similar  argument ;  only  one  distinction  occurs, 
namely,  in  this  case,  contrary  to  the  other,  the  curve  AIFI 
is  concave  to  the  axis  AD. 

COR.  It  should  be  noted  that  DE. DT  =  R.  DF  =  area 
ADEZ.* 

12.  From   the  preceding  we  can   deduce  the  following 
theorem. 

Let  ZGE,  AKF  be  any  two  lines  so  related  that,  if  any 
straight  line  EOF  is  applied  to  a  common  axis  AD,  the 
square  on  DF  is  always  equal  to  twice  the  space  ADEZ; 
also  take  DQ,  along  AD  produced,  equal  to  DE,  and  join 
FQ;  then  FQ  is  perpendicular  to  the  curve  AKF. 

I  will  also  add  the  following  kindred  theorems. 

13.  Let  AGEZ  be  any  curve,  and  D  a  certain  fixed  point 
such  that  the  radii,  DA,  DG,   DE,  drawn  from  D,  decrease 
continually  from  the  initial  radius   DA;   then  let  DKE  be 
another  curve  intersecting  the   first  in    E   and  such  that, 
if  any  straight  line  DKG  is  drawn   through  ;D,  cutting  the 
curve  AEZ  in  G  and  the  curve   DKE  in   K,   the  rectangle 
contained  by  DK  and  a  given  length  R  is  equal  to  the  area 
ADG ;  also  let  DT  be  drawn  perpendicular  to  DE,  so  that 
DT  =  2R ;  join  TE.     Then  TE  touches  the  curve  DKE. 

Moreover,  if  any  point,  K  say,  is  taken  in  the  curve  DKE, 
and  through  it  DKG  is  drawn,  and  DG  :  DK  -  R  :  P ;  then,  if 
DT  is  taken  equal  to  2P  and  TG  is  joined,  and  also  KS  is 
drawn  parallel  to  GT;  KS  will  touch  the  curve  DKE. 

*  See  note  at  end  of  this  lecture. 


LECTURE  X  119 

Observe  that     Sq.  on  DG  :  Sq.  on  DK  =  2R  :  DS. 

Now,  the  above  theorem  is  true,  and  can  be  proved  in 
a  similar  way,  even  if  the  radii  drawn  from  D,  DA,  DG,  DE, 
are  equal  (in  which  case  the  curve  AGEZ  is  a  circle  and 
the  curve  DKE  is  the  Spiral  of  Archimedes),  or  if  they  con- 
tinually increase  from  A. 

14.  From  this  we  may  easily  deduce  the  following 
theorem. 

Let  AGE,  DKE  be  two  curves  so  related  that,  if  straight 
lines  DA,  DG  are  drawn  from  some  fixed  point  D  in  the 
curve  DKE  (of  which  the  latter  cuts  the  curve  DKE  in  K), 
the  square  on  DK  is  equal  to  four  times  the  area  ADG  ;  draw 
DH  perpendicular  to  DG,  and  make  DK  :  DG  =  DG  :  DH  ;  join 
HK;  then  HK  is  perpendicular  to  the  curve  DKE. 


We  have  now  finished  in  some  fashion  the  first  part,  as 
we  declared,  of  our  subject.  Supplementary  to  this  we 
add,  in  the  form  of  appendices,  a  method  for  finding 
tangents  by  calculation  frequently  used  by  us  (a  nobis 
usitatum).  Although  I  hardly  know,  after  so  many  well- 
known  and  well-worn  methods  of  the  kind  above,  whether 
there  is  any  advantage  in  doing  so.  Yet  I  do  so  on  the 
advice  of  a  friend •  and  all  the  more  willingly,  because  it 
seems  to  be  more  profitable  and  general  than  those  which 
I  have  discussed.* 

*  See  note  at  the  end  of  this  lecture. 


M 


R 


120  BARROW'S  GEOMETRICAL  LECTURES 

Let  AP,  PM  be  two  straight  lines  given  in  position,  of 
which  PM  cuts  a  given  curve  in  M,  and  let  MT  be  supposed 
to  touch  the  curve  at  M,  and  to  cut  the  straight  line  at  T. 

In  order  to  find  the  quantity  of  the  straight  line  PT,* 
I  set  off  an  indefinitely  small  arc,  MN,  of  / 

the  curve;  then  I  draw  NQ,  NR  parallel  to 
MP,  AP;  I  call  MP  =  mt  PT  =  /,  MR  =  a, 
N  R  =  £,  and  other  straight  lines,  determined 
by  the  special  nature  of  the  curve,  useful  A     *r    Q 
for  the  matter  in  hand,  I  also  designate         Fig.  115. 
by  name;    also  I  compare   MR,    NR  (and  through  them, 
MP,  PT)  with  one  another  by  means  of  an  equation  obtained 
by  calculation ;  meantime  observing  the  following  rules. 

RULE  i.  In  the  calculation,  I  omit  all  terms  containing 
a  power  of  a  or  e>  or  products  of  these  (for  these  terms 
have  no  value). 

RULE  2.  After  the  equation  has  been  formed,  I  reject 
all  terms  consisting  of  letters  denoting  known  or  deter- 
mined quantities,  or  terms  which  do  not  contain  a  or  e 
(for  these  terms,  brought  over  to  one  side  of  the  equation, 
will  always  be  equal  to  zero). 

RULE  3.  I  substitute  m  (or  MP)  for  a,  and  /  (or  PT)  for 
e.  Hence  at  length  the  quantity  of  PT  is  found. 

Moreover,  if  any  indefinitely  small  arc  of  the  curve  enters 
the  calculation,  an  indefinitely  small  part  of  the  tangent, 
or  of  any  straight  line  equivalent  to  it  (on  account  of  the 

*  See  note  at  the  end  of  this  lecture. 


LECTURE  X  121 

indefinitely  small  size  of  the  arc)  is  substituted  for  the  arc. 
But  these  points  will  be  made  clearer  by  the  following 
examples. 


NOTE 

Barrow  gives  five  examples  of  this,  the  "differential 
triangle  "  method.  As  might  be  expected,  two  of  these  are 
well-known  curves,  namely  the  Folium  of  Descartes,  called 
by  Barrow  La  Galandc,  and  the  Quadratrix ;  a  third  is  the 
general  case  of  the  quasi-circular  curves  xn+yn  =  an;  the 
fourth  and  fifth  are  the  allied  curves  r  =  a .  tan  0  and 
y  =  a  .  tan  x.  It  is  noteworthy,  in  connection  with  my  sug- 
gestion that  Barrow  used  calculus  methods  to  obtain  his 
geometrical  constructions,  that  he  has  already  given  a  purely 
geometrical  construction  for  the  curve  r  =  a  .  tan  0  in  Lect. 
VIII,  §  1 8,  if  the  given  lines  are  supposed  to  be  at  right  angles. 
I  believe  that  Barrow,  by  including  this  example,  intends  to 
give  a  hint  as  to  how  he  made  out  his  geometrical  con- 
struction :  thus  : — 

The  equation  of  the  curve  is  x*  +  x2y2  =  a2y2 -,  the 
gradient,  as  he  shows  is  x(2X2  +y2)/y(a2  -  x2) ;  using  the 
general  letters  x  and  y  instead  of  his  p  and  m.  Descartes 
has  shown  that  a  hyperbola  is  a  curve  having  an  equation 
of  the  second  degree,  hence  Barrow  knows  that  its  gradient 
is  the  quotient  of  two  linear  expressions,  and  finds  (?  by 
equating  coefficients)  the  hyperbola  whose  gradient  is 
xQ(2XXQ+yyQ)/y0(az - xx^) ;  the  feasibility  of  this  is  greatly 
enhanced  by  the  fact  that  Barrow  would  have  written  the 
two  gradients  as 

m:t  =  x(2xx+yy)  :y(aa  - xx)   and 

x0(2xx0  +yy0)  '.y^aa  •=•-  xx0).      . 

These  two  gradients  are  the  same  at  the  point  XQ,  >'0;  hence 
if  he  can  find  such  a  hyperbola,  it  will  touch  the  curve ;  he 
can  draw  its  tangent,  and  this  will  also  be  a  tangent  to 
his  curve.  The  curve  does  turn  out  to  be  a  hyperbola; 
for  its  equation  is  x2x2Q  +  X^Q  .  xy  =  a2yyQ  or  x2  +y2  = 
))  •  (d~x)]>  where  d  =•  a2/x.  This,  latter  form  is 


122  BARROW'S  GEOMETRICAL  LECTURES 

easily  seen  to  be  equivalent  to  the  construction,  in  Lect.  VIII, 
§  1 7,  for  the  curve  DYO,  when  the  axes  are  rectangular ;  for 
the  equation  gives  DY2  =  YE  .  YR.  It  also  suggests  that  the 
construction  for  the  original  curve  is  transformable  into 
that  of  §  17,  as  is  proved  by  Barrow  in  §  18,  and  in  order 
that  Barrow  may  draw  the  tangent,  §§  10,  1 1,  12  of  Lect.  VI 
are  necessary  to  prove  that  the  auxiliary  is  a  hyperbola  of 
which  the  asymptotes  can  be  determined  by  a  fairly  easy 
geometrical  construction.  Barrow  then  generalizes  his 
theorems  for  oblique  axes.  I  contend  that  this  suggestion 
is  a  very  probable  one  for  three  reasons :  (i)  it  is  quite 
feasible,  even  if  it  is  considered  to  be  far-fetched,  (ii)  we 
know  that  mathematicians  of  this  time  were  jealous  of  their 
methods,  and  gave  cryptogrammatic  hints  only  in  their  work 
(cf.  Newton's  anagram),  and  (iii)  it  is  to  my  mind  the  only 
reason  why  this  particular  theorem  should  have  been  selected 
(especially  as  Barrow  makes  it  his  Example  i),  for  there  is 
no  great  intrinsic  worth  in  it. 

The  fifth  example,  the  case  of  the  curve  y  =  a  .  tan  0,  I 
have  selected  for  giving  in  full,  for  several  reasons.  It  is 
the  clearest  and  least  tedious  example  of  the  method,  it 
is  illustrated  by  two  diagrams,  one  being  derived  from  the 
other,  and  therefore  the  demonstration  is  less  confused,  it 
is  connected  with  the  one  discussed  above  and  suggests 
that  Barrow  was  aware  of  the  analogy  of  the  differential 
form  of  the  polar  subtangent  with  the  Cartesian  subtangent, 
and  that  in  this  is  to  be  found  the  reason  why  Barrow 
gives,  as  a  rule,  the  polar  forms  of  all  his  Cartesian  theorems  ; 
and  lastly,  and  more  particularly,  for  its  own  intrinsic 
merits,  as  stated  below.  Barrow's  enunciation  and  proof 
are  as  follows  : — 


EXAMPLE  5.  Let  DEB  be  a  quadrant  of  a  circle,  to 
which  BX  is  a  tangent;  then  let  the  line  AMO  be  such 
that,  if  in  the  straight  line  AV  any  part  AP  is  taken  equal 
to  the  arc  BE,  and  PM  is  erected  perpendicular  to  AV,  then 
PM  is  equal  to  BG  the  tangent  of  the  arc  BE. 


LECTURE  X 


123 


Take  the  arc  BF  equal  to  AQ  and  draw  CFH  ;  drop  EK, 
FL  perpendicular  to  CB.     Let  CB  =  r,  CK  =/  KE  =  g. 
x 


K  L  A     T    Q  P 

Fig.  120.  Fig.  121. 

Then,  since  CE  :  EK  =  arc  EF  :  LK  =  QP  :  LK ;  therefore 
r:g  =  e  :  LK,  or  LK  =  ge/r,  and  CL  =  f+gelr;  hence  also 
LF  -  J(r*  ~f2  ~  *falr)  =  J(g2  -  tfgelr). 

But  CL  :  LF  =  CB  :  BH,or/+£*/r:  J(g2  -  2fgefr)  =  r:  m  -a, 
and  squaring,  we  have 

f2  +  2-fgelr  '•  S2  ~  2fSelr  —  r2:m2-  2 ma. 
Hence,  omitting  the  proper  terms,  we  obtain  the  equation 

rfma  =  gr2e  +  gm2e '} 
and,  on  substituting  mt  /  for  a,  e,  we  get 

rfm*  =  gr*t+gm*t,     or     rfm*l(gr*+gm*)  =  t. 
Hence,  since  m  =  rgjf,  we  obtain 
/  -  m.r*l(f*  +  m*)  =  BG.CB2/CG2  =  BG.CK2/CE2. 

In  other  words,  this  theorem  states  that,  if  y  =  tan  x, 
where  x  is  the  circular  measure  of  an  "angle"  or  an  "arc," 
then  dy\dx  =  m\t  =  CE2/CK2  -  sec2  x. 

Moreover,  although  Barrow  does  not  mention  the  fact, 
he  must  have  known  (for  it  is  so  self-evident)  that  the  same 
two  diagrams  can  be  used  for  any  of  the  trigonometrical 
ratios.  Therefore  Barrow  must  be  credited  with  the  differ- 
entiation of  the  circular  functions.  (See  Note  to  §  15  of 
App.  2  of  Lect.  XII.) 


124  BARROW'S  GEOMETRICAL  LECTURES 

As  regards  this  lecture,  it  only  remains  to  remark  on  the 
fact  that  the  theorem  of  §  1 1  is  a  rigorous  proof  that 
differentiation  and  integration  are  inverse  operations,  where 
integration  is  defined  as  a  summation.  Barrow  not  only, 
as  is  well  known,  was  the  first  to  recognise  this ;  but  also, 
judging  from  the  fact  that  he  gives  a  very  careful  and  full 
proof  (he  also  gave  a  second  figure  for  the  case  in  which 
the  ordinates  continually  decrease),  and  in  addition,  as  will 
be  seen  in  Lect.  XI,  §  19,  he  takes  the  trouble  to  prove  the 
theorem  conversely, — judging  from  these  facts,  I  say, — he 
must  have  recognised  the  importance  of  the  theorem  also. 
It  does  not  seem,  however,  to  have  been  remarked  that  he 
ever  made  any  use  of  this  theorem.  He,  however,  does 
use  it  to  prove  formulae  for  the  centre  of  gravity  and  the 
area  of  a  paraboliform,  which  formulae  he  only  quotes  with 
the  remark,  "of  which  the  proofs  may  be  deduced  in 
various  ways  from  what  has  already  been  shown,  without 
much  difficulty  "  (see  note  to  Lect.  XI,  §  2). 

The  "differential  triangle"  method  has  already  been 
referred  to  in  the  Introduction ;  it  only  remains  to  point 
out  the  significance  of  certain  words  and  phrases.  Barrow, 
whilst  he  acknowledges  that  the  method  "  seems  to  be 
more  profitable  and  more  general  than  those  which  I  have 
discussed,"  yet  is  in  some  doubt  as  to  the  advantage  of 
including  it,  and  almost  apologizes  for  its  insertion ; 
probably,  as  I  suggested,  because,  although  he  has  found 
it  a  most  useful  tool  for  hinting  at  possible  geometrical 
constructions,  yet  he  compares  it  unfavourably  as  a  method 
with  the  methods  of  pure  geometry.  It  is  also  to  be 
observed  that  his  axes  are  not  necessarily  rectangular, 
although  in  the  case  of  oblique  axes,  PT  can  hardly  be 
accepted  as  the  subtangent ;  hence  he  finds  it  convenient 
tto  tacitly  assume  that  his  axes  are  at  right  angles.  The 
last  point  is  that  Barrow  distinctly  states  that  his  method 
is  expressly  "in  order  to  find  the  quantity  of  the  sub- 
tangent,"  and  I  consider  that  this  is  almost  tantamount  to 
a  direct  assertion  that  he  has  used  it  frequently  to  get  his 
first  hint  for  a  construction  in  one  of  his  problems.  The 
final  significance  of  the  method  is  that  by  it  he  can  readily 
handle  implicit  functions. 


LECTURE    XI 


Change  of  the  independent  variable  in  integration.  Integra- 
tion  the  inverse  of  differentiation.  Differentiation  of  a  quotient. 
Area  and  centre  of  gravity  of  a  paraboliform.  Limits  for  the 
arc  of  a  circle  and  a  hyperbola.  Estimation  of  IT. 

NOTE 

In  the  following  theorems,  Barrow  uses  his  variation  of 
the  usual  method  of  summation  for  the  determination  of  an 
area.  If  ABKJ  is  the  area  under  the  curve  AJ,  he  divides 
BK  into  an  infinite  number  of  equal  parts  and  erects 
ordinates.  In  his  figures  he  generally  makes  four  parts 
do  duty  for  the  infinite  number. 

He  then  uses  the  notation  already 
mentioned,  namely,  that  the  area  ABKJ 
is  equal  to  the  sum  of  the  ordinates  AB, 
CD,  EF,  GH,  JK. 

The  same  idea  is  involved  when  he 
speaks  of  the  sum  of  the  rectangles 
CD.  DB,  EF.  FD,  GH  .  FH,  JK  .  GH  ; 
for  this  sum,  where  commas  are  used 
between  the  quantities  instead  of  a  plus  B  D  F  H  K 
sign,  does  not  stand  for  the  area  ABKJ,  but  for  R  .  A'BKJ', 
where  an  ordinate  HG'  is  such  that  R  .  HG'  =  HG  .  FH, 
and  R  is  some  given  length ;  in  other  words,  ordinates 
proportional  to  each  of  the  rectangles  are  applied  to  points 
of  the  line  BK,  and  their  aggregate  or  sum  is  found ; 
hence  this  sum  is  of  three  dimensions.  On  the  contrary, 
he  uses  the  same  phrase,  with  plus  signs  instead  of  commas, 
to  stand  for  a  simple  summation. 


126  BARROW'S  GEOMETRICAL  LECTURES 

Thus,  in  §  3  of  this  lecture,  the  sum  of  AZ  .  AE2,  BZ  .  BF2, 
CZ .  CG2,  etc.,  is  the  area  aggregated  from  ordinates 
proportional  to  AZ.AE2,  BZ.BF2,  CZ .  CG2,  etc.,  applied 
to  the  line  YD ;  and  it  is  of  the  fourth  dimension. 
Whereas,  in  §  3,  the  sum  HL .  H02+ LK  .  LY2+ Kl .  KY2  + 
etc.,  is  aggregated  from  ordinates  equal  to  HO2,  LY2,  KY2, 
etc.,  applied  to  the  line  HD;  and  it  is  the  same  as  the 
sum  of  HO2,  LY2,  KY2,  etc. 


i.  If  VH  is  a  curve  whose  axis  is  VD,  and  HD  is  an 
ordinate  perpendicular  to  VD,  and  <j>Z\J/  is  a  line  such  that, 
if  from  any  point  chosen  at  random  on  the  curve,  E  say, 
a  straight  line  EP  is  drawn  normal  to  the  curve,  and  a 
straight  line  EAZ  perpendicular  to  the  axis,  AZ  is  equal  to 
the  intercept  AP;  then  the  area  VDi/^  will  be  equal  to  half 
the  square  on  the  line  DH. 

For  if  the  angle  HDO  is  half  a  right  angle,  and  the 
straight  line  VD  is  divided  into  an  infinite  number  of 
equal  parts  at  A,  B,  C,  and  if  through  these  points  straight 
lines  EAZ,  FBZ,  GCZ,  are  drawn 
parallel  to  HD,  meeting  the  curve 
in  E,  F,  G ;  and  if  from  these 
points  are  drawn  straight  lines  EIY, 
FKY,  GLY,  parallel  to  VD  or  HO; 
and  if  also  EP,  FP,  GP,  HP  are 
normals  to  the  curve,  the  lines 
intersecting  as  in  the  figure ;  then 
the  triangle  HLG  is  similar  to  the  Fig-  I22- 

triangle  PDH  (for,  on  account  of  the  infinite  section,  the 
small  arc  HG  can  be  considered  as  a  straight  line). 


E 
F/ 

V^ 

4 

/*  — 

A 

zl 

\ 

B 

z/ 

G/\ 

N\ 

C 

Z/ 

/j\M 

\  \ 

D 

& 

y 

/Y 
0 

J  \/* 

P 

P 

P 
P 

LECTURE  XI  127 

Hence,     HL:LG  =  PD:DH,     or     HL-DH  =  LG.PD, 
i.e.  HL.HO  =  DC .  D^. 

By  similar  reasoning  it  may  be  shown  that,  since  the 
triangle  GMF  is  similar  to  the  triangle  PCG,  LK  .  LY  =  CB  .  CZ; 
and  in  the  same  way,  Kl  .  KY  =  BA  .  BZ,  ID  .  IY  =  AV  .  AZ. 

Hence  it  follows  that  the  triangle  DHO  (which  differs  in 
the  slightest  degree  only  from  the  sum  of  the  rectangles 
HL  .  HO  +  LK  .  LY  +  Kl  .  KY  +  ID  .  IY)  is  equal  to  the  space 
VD\f/<f>  (which  similarly  differs  in  the  least  degree  only  from 
the  sum  of  the  rectangles  DC  .  D«A  +  CB  .  CZ+  BA  .  BZ 

+  AV.AZ); 

i.e.  DH2/2  =  space  VO^. 

A  lengthier  indirect  argument  may  be  used;  but  what 
advantage  is  there? 

2.  With  the  same  data  and  construction  as  before,  the 
sum  of  the  rectangles  AZ  .  AE,  BZ  .  BF,  CZ  .  CG,  etc.,  is  equal 
to  one-third  of  the  cube  on  the  base  DH. 

For,  since  HL :  LG  =  PD  :  DH  =  PD  .  DH  :  DH2;  therefore 
HL.DH2=LG.PD.DH  or  LH  .  H02  =  DC  .  D^  .  DH  ;  and, 
similarly  LK .  LY2  =  CB  .  CZ .  CG,  Kl .  KY2  =  BA  .  BZ .  BF,  etc. 

ButthesumHL.H02+LK.LY2+KI.KY2  +  etc.  =  DH3/3;* 
and  the  proposition  follows  at  once. 

3.  By  similar  reasoning,  it  follows  that 

the  sum  of  AZ  AE2,  BZ .  BF2,  CZ  .  CG2,  etc.  =  DH4/* ; 
the  sum  of  AZ  .  AE3,  BZ  .  BF3,  CZ  .  CG3,  etc.  =  DH5/5  ; 
and  so  on.f 


*  See  the  critical  note  immediately  following. 

f  The  analytical  equivalents  of  the  theorems  given  above  are  comprised 
in  the  general  formula  (with  their  proofs), 

fyr(dyldX).  dx=ff.  </ 


128  BARROWS  GEOMETRICAL  LECTURES 


NOTE 
On  the  assumptions  in  the  proofs  of  §§  2,  3. 

The  summation  used  in  §  2  has  already  been  given  by 
Barrow  in  Lect.  IV ;  he  states  that  it  has  been  established 
"in  another  place"  (?  by  Wallis  or  others),  and  that  it  at 
least  " is  sufficiently  known  among  geometers" 

It  is  easy,  however,  to  give  a  demonstration  according 
to  Barrow's  methods  of  the  general  case ;  and,  since  in 
several  cases  Barrow  is  content  with  saying  that  the  proof 
may  easily  be  obtained  by  his  method,  and  sometimes  he 
adds  "in  several  different  ways,"  I  feel  sure  that  he  had 
made  out  a  proof  for  these  summations  in  the  general  case. 

The  method  given  below  follows  the  idea  of  Lect.  IV, 
by  finding  a  curve  convenient  for  the  summation,  without 
proving  that  this  curve  is  the  only  one  that  will  do.  Other 
methods  will  be  given  later,  thus  substantiating  Barrow's 
statement  that  the  matter  may  be  proved  in  several  ways  ; 
see  notes  following  Lect.  XI,  §  27,  and  Lect.  XI,  App.,  §  2. 

Let  AH,  KHO  be  two  straight  lines 
at  right  angles,  and  let  AH  =  HO  =  R 
and  KH  =  8.  Let  AEO  be  a  curve  such 
that  (in  the  figure)  UE  is  the  first  of  A 
n-i  geometrical  means  between  UW 
and  UV, 

i.e.  UEn  =  UW-MJY, 

or  DY"  =  ADn  =  R*-1.  DE. 

Let  AFK  be  a  curve  such  that  PF  is 
the  first  of  n  geometrical  means  between 
PQ  and  PL,  or  PFM+1  =  PQn .  PL,  i.e.  8  .  ADW+1  =  Rn+1 .  DF. 

Then  the  curve  AFK  is  a  curve  that  is  fitted  for  the 
determination  of  the  area  under  the  curve  AED,  providing 
a  suitable  value  of  S/R  is  chosen. 


and  if  8  is  taken  equal  to  R/(»  +  i),  FD  :  DT  =  DE  :  R ;  and 

therefore  by  Lect.  X,  §  n, 

the  sum  DY".  DD'  + .  .  .  =  the  sum  Rn~l .  DE  .  DD' 

-  R"-1 .  area  ADE 

=  R".  DF  =  S.  AD"+1/R  =  DY"+1/(«  +  i ). 


LECTURE  XI  129 

Hence     we     may     deduce     the     following     important 
theorems  :  — 


4.  Let  VDi/f<£  be  any  space  of  which  the  axis  YD  is 
equally  divided  (as  in  fig.  122);  then  if  we  imagine 
that  each  of  the  spaces  VAZ</>,  VBZ<£,  VCZ</>,  etc.,  is 
multiplied  by  its  own  ordinate  AZ,  BZ,  CZ,  etc.,  respectively, 
the  sum  which  is  produced  will  be  equal  to  half  the  square 
of  the  space 


5.  If,  however,  each  of  the  square  roots  of  the  spaces  is 
multiplied  by  its  own  ordinate,  an  aggregate  is  produced 
equal  to  two-thirds  of  the  square  root  of  the  cube  of 


6.  Example.  —  Let  VDi/f  be  a  quadrant  of  a  circle,  of 
which  the  radius  is  R  and  the  perimeter  is  P;  then  the 
segments  VAZ,  VBZ,  VCZ,  .  .  .  ,  each  multiplied  by  its 
own  sine,  AZ,  BZ,  CZ,  .  .  .  ,  respectively,  will  together 
make  R2P2/8. 

Also  the  sum  AZ  .      VAZ  +  BZ  .      VBZ  +  etc. 


Again,  if  VDi/f  is  a  segment  of  a  parabola,  the  sum  made 
from  the  products  into  the  ordinates  will  be  equal  to  2/9 
of  YD2  .  Dip,  and  that  from  the  products  of  the  square  roots 
of  the  segments  into  the  ordinates  will  be  equal  to  2/3  of 
V(8/27  of  YD*  .  D^)  or  ^(VD3  .  Dip  .  3  2/243). 


*  The  equivalents  of  §§  4,  5,  are  respectively  //(/y  .  dx)  .  dx  —  (  fy  .  dx}z 
andyVV(y)/  .  dx)  .  dx  =  (fy  .  dx)3/2.  2/3  ;  or  in  a  more  recognizable  form( 
putting  •£  tor  fy  .  dx,  they  a.refz(dzldx)  .  dx  =  zz/2,  and  so  on. 

9 


130  13 ARROWS  GEOMETRICAL  LECTURES 


Other  similar  things  concerning  the  sums  made  from  the 
products  of  other  powers  and  roots  of  the  segments  into  the 
ordinates  or  sines  can  be  obtained. 

7.  Further,  it  follows  from  what  has  gone  before  that,  in 
every  case,  if  the  lines  VP  intercepted  between  the  vertex 
and  the  perpendiculars  are  supposed  to  be  applied  through 
the  respective  points  A,  B,  C,  .  .  .  ,  say  that  AY,  BY,  CY,  .  .  . 
are  equal  to  the  respective  lines  VP;  then  will  the  space 
VD£<9,  constituted  by  these  applied  lines,  be  equal  to  half 
the  square  on  the  subtense  VH. 

8.  Moreover,  if  with  the  same  data,    RXXS   is   a   curve 
such  that  IX  =  AP,  KX  =  BP,  LX  =  CP,  .  .  .  ;  then  the  solid 
formed  by  the  rotation  of  the  space  VDi/^  about  VD  as  an 
axis  is  half  the  solid  formed  by  the  rotation  of  the  space 
DRSH  about  the  same  axis  VD. 

9.  All  the  foregoing  theorems  are  true,  and  for  similar 
reasons,  even  if  the  curve  VEH  is  convex  to  the  line  VD. 

From  these  theorems,  the  dimensions  of  a  truly  bound- 
less number  of  magnitudes  (proceeding  directly  from  their 
construction)  may  be  observed,  and 
easily  verified  by  trial. 


10.  Again,  if  VH  is  a  curve,  whose 
axis  is  VD  and  base  DH,  and  DZZ  is 
a  curve  such  that,  if  any  point  such 
as  E  is  taken  on  the  curve  VH  and 
ET  is  drawn  to  touch  the  curve,  and 
a  straight  line  EIZ  is  drawn  parallel  to 


Fig.  125. 


LECTURE  XI  131 

the   axis,  then    IZ  is    always   equal    to    AT;   in  that  case, 
I  say,  the  space  DHO  is  equal  to  the  space  VHD. 

This  extremely  useful  theorem  is  due  to  that  most  learned 
man,  Gregory  of  Aberdeen :  *  we  will  add  some  deductions 
from  it. 

1 1.  With  the  same  data,  the  solid  formed  by  the  rotation 
of  the   space    DHO  about  the  axis    VD  is  twice  the  solid 
formed  by  the  rotation  of  the  space  VDH  about  the  same 
axis. 

For     HL :  LG  =  DH  :  DT  =  DH  :  HO  =  DH2 :  DH  .  HO ; 
HL.DH.HO  =  LG.DH2  =  CD.  DH2. 

Similarly,  LK.DL.  LZ  =  BC.CG2,  Kl.  DK.  KZ  =  AB.BF2, 
and  ID.DI.IZ  =  VA  .  AE2. 

But  it  is  well  known  that  t 
the  sum    CD .  DH2  +  BC  .  CG2  +  AB  .  BF2  +  VA .  AE2 

=  twice  the  sum  of  Dl .  IE,  DK  .  KF,  DL.  LG,  etc., 
and  therefore  the  solid  formed  by  the  space  DHOjrotated 
about  the  axis  VD  is  double  of  the  solid  formed  by  the 
space  VDH  rotated  round  VD. 

12.  Hence  the  sum  of  Dl  .  IZ,  DK.  KZ,  DL  .  LZ,  etc.  (ap- 
lied  to  HD)  =  the  sum  of  the  squares  on  the  ordinates  to  VD, 

=  the  sum  of  AE2,  BF2,  CG2,  etc.  (applied  to  VD). 
The  same  even  tenor  of  conclusions  is  observable  for  the 
other  powers. 

*  The  member  of  a  remarkable  family  of  mathematicians  and  scientists 
that  is  here  referred  to  is  James  Gregory  (1638-1675),  who  published  at 
Padua,  in  1668,  Geometries  Pars  Univer salts.  He  also  gave  a  method  for 
infinitely  converging  series  for  the  areas  of  the  circle  and  hyperbola  in  1667. 

J-  For  a  discussion  of  this  and  §§  12,  13,  14,  see  the  critical  note  on 
Page  133- 


132  BARROW'S  GEOMETRICAL  LECTURES 

13.  By  similar  reasoning,  it  follows  that 

the  sum  of  Dl2 .  IZ,  DK2 .  KZ,  DL2 .  LZ,  etc.  (applied  to  HD) 
=  three  times  the  sum  of  the  cubes  on  all  the  ordi- 
nates  AE,  BF,  CG,  .  .  .,  applied  to  YD. 

14.  With  the  same  data,  if  DXH  is  a  curve  such  that  any 
ordinate  to  DH,  as  IX,  is  a  mean  proportional  between  the 
ordinates  IE,  IZ  congruent  to  it  j  then  the  solid  formed  by 
the  space  VDH  rotated  about  the  axis  DH  is  double  the 
solid  formed  from  the  space  DXH  rotated  about  the  same 
axis. 

15.  If,  however  (in  fig.  125),  the  curve  DXH  is  supposed 
to   be   such   that  any  ordinate,    CX    say,    is   a    bimedian  * 
between  the  congruent  ordinates  IE,  IZ  ;  then  the  sum  of 
the  cubes  of  IX,  KX,  LX,  etc.,  is  one-third  of  the  sum  of  the 
cubes  of  DV,  IE,  KF,  etc.     But  if  IX  is  a  trimedian*;  then 
the  sum  of  IX4,  KX4,  LX4,  etc.,  is  equal  to  one-fourth  of  the 
sum  of  DV4,   IE4,   KF4,   etc.;   and   so   on   for  all  the  other 
powers. 

*  NOTE.  I  call  by  the  name  bimedian  the  first  of  two 
mean  proportionals,  by  trimedian  the  first  of  three,  and 
so  on. 

These  results  are  deduced  and  proved  by  similar  reason- 
ing to  that  of  the  previous  propositions ;  but  repetition  is 
annoying,  t 

1 6.  Again,  if  VYQ  is  a  line  such  that  the  ordinate  AY  is 
equal  to  AT^  BY  to  BT,  and  so  on  \  then  the  sum  of  IZ2,  KZ2, 
LZ2,  etc  ,  that  is,  the  sum  of  the  squares  of  the  ordinates 


f  Literally,  "  it  irks  me  to  cry  cuckoo." 


LECTURE  XI  133 

of  the  curve  DZO  applied  to  the  line  DH,  is  equal  to  the 
sum  VA  .  AE .  AY  +  AB  .  BF .  BY  +  etc.,  that  is,  the  figure  VDH 
"multiplied"  by  the  figure  VDQ.* 

17.  Also     the  sum  of  IZ3,  KZ3,  LZ3,  etc. 

m  the  sum  VA.AE.AY2  +  AB.BF.BY2  +  etc.  ;* 
that  is,  the  figure  VDH  "multiplied"  by  the  figure  VDQ 
"squared." 

These  you  can  easily  prove  by  the  pattern  of  the  proofs 
given  above. 

1 8.  The   same   things   are   true  and  are  proved  in  an. 
exactly  similar  manner,  even  if  the  curve  VH  is  convex  to 
the  straight  line  VD. 

NOTE 

The  equality,  which  in  §  1 1  is  said  by  Barrow  to  be  well 
known,  namely,  the  sum  CD  .  DH2  +  BC  .  CG2  +  etc. 

=  twice  the  sum  of  Dl  .  IE,  DK .  KF,  etc  , 
is  really  an  equality  between  two  expressions  for  the  volume 
of  the  solid  formed  by  the  rotation  of  VDH  about  VD;  and 
the  analytical  equivalent  is  JJF2  dx  =  ^^xy  dy,  with  the  inter- 
mediate step  z^ydydx.  Thus  these  theorems  of  Barrow 
are  equivalent  to  the  equality  of  the  results  obtained  from 
a  double  integral,  when  the  two  first  integrals  are  obtained 
by  integrating  with  regard  to  each  of  the  two  variables  in 
turn.  He  says  indeed  that  the  first  result,  that  in  §  n,  is 
a  matter  of  common  knowledge,  but  he  remarks  that  the 
others  that  he  uses  in  the  following  sections  can  be  obtained 
by  similar  reasoning.  From  this,  and  from  indications  in 
Lect.  IV,  §  1 6,  Lect.  X,  §  1 1,  and  §  19  of  this  lecture,  I  feel 

*  The  equivalents  of  these  theorems  are  : 
16. 


134  BARROW'S  GEOMETRICAL  LECTURES 

certain  that  Barrow  had  obtained  these  theorems  in  the 
course  of  his  researches,  but,  as  in  many  other  cases,  he 
omits  the  proofs  and  leaves  them  to  the  reader.  All  the 
more,  because  the  proofs  follow  very  easily  by  his  methods. 
Take,  for  the  sake  of  example,  the  case  of  the  equivalent  of 
JJJF3  dy  dx,  for  which  I  imagine  Barrow's  method  would  have 
been  somewhat  as  follows. 

In  order  to  find  the  aggregate  of  all  the  points  of  a  given 
space  VDH,  each  multiplied  by  the  cube  of  its  distance  from 
the  axis  VD,  we  may  proceed  in  two  different  ways. 

Method  i . — By  applying  lines  PX,  proportional  to  the 
cube  of  BP,  to  every  point  P  of  the  line  BF,  find  the 


X 

/ 

«• 

N, 

—  1 

F. 

A 

/ 

f 

f 

> 

\ 

K    LHX 

aggregate  of  the  products  for  the  line  BF;  then  find  the  sum 
of  these  aggregates  for  all  the  lines  applied  to  VD.  Here, 
since  PX  is  proportional  to  BP3,  the  curve  BXX  is  a  cubical 
parabola,  and  the  space  BFX  is  one-fourth  of  the  fourth 
power  of  BF  ;  and  Barrow  would  write  the  result  of  the 
summation  as  the  sum  of  AE4/4,  BF4/4,  CG4/4,  etc.  (applied 
to  VD). 

Method  2. — Find  the  aggregates  along  all  the  parallels 
to  VD,  and  then  the  sum  of  these  aggregates  applied  to  DH- 
Here  the  first  aggregates  are  represented  by  rectangles  whose 
bases  are  IE,  KF,  etc.,  and  whose  heights  are  equal  to  Dl3, 
DK3,  etc.,  respectively  ;  and  Barrow  would  write  this  as  the 
sum  of  Dl3 .  IE,  DK3 .  KF,  etc.  (applied  to  DH). 

Lastly,  in  fig.  125,  ID.DIMZ  =  VA  .  AE3  .  Dl,  etc. ;  hence 
all  the  results  follow  immediately;  i.e.  fy^dxdy  is  equal 
to  either  of  the  integrals  frl^dx,  \xy*  dy ;  and  Barrow 
proves  that  these  are  equal  to  one-fourth  of  ^yKdyjdx)  dy. 


LECTURE  XI 


35 


K 

Vz'_ 

xz 

^, 

L 

T       A 

G 

F 

«._.  R-_^ 

V 

D            Pj 

\ 

\ 

N 

O 

s    J 

\ 

/ 

MX^ 

E    /Y 

B  "Q 

19.  Again,  let  AM B  be  a  curve  of  which  the  axis  is  AD 
and  let  BD  be  perpendicular  to  AD ;  also  let  KZL  be  another 
line  such  that,  when  any  point  M  is  taken  in  the  curve  AB, 
and  through  it  are  drawn  MT  a  tangent  to  the  curve  AB, 
and  MFZ  parallel  to  DB,  cutting  KZ  in  Z  and  AD  in  F,  and 
R  is  a  line  of  given  length,  TF :  FM  =  R  :  FZ.     Then  the 
space   ADLK    is    equal   to    the   rectangle    contained    by    R 
and  DB.* 

For,  if  DH  =  R  and  the  rect- 
angle BDHI  is  completed,  and 
MN  is  taken  to  be  an  indefinitely 
small  arc  of  the  curve  AB,  and 
MEX,  N08  are  drawn  parallel  to 
AD  ;  then  we  have 

NO:  MO  =  TF:FM  =  R:FZ;  Fig'  I27' 

.-.  NO.FZ  =  MO.R,     and     FG.FZ=ES.EX. 
Hence,  since  the  sum  of  such  rectangles  as  FG  .  FZ  differs 
only  in   the  least   degree  from   the  space  ADLK,   and  the 
rectangles  ES .  EX  form  the  rectangle  DHIB,  the  theorem  is 
quite  obvious. 

20.  With  the  same  data,  if  the  curve  PYQ  is  such  that 
the  ordinate  EY  along  any  line  MX  is  equal  to  the  corre- 
sponding FZ ;  then  the  sum  of  the  squares  on  FZ  (applied 
to    the   line  AD)   is    equal    to    the    product   of  R   and  the 
space  DPQB. 

21.  Similarly,  the  sum  of  the  cubes  of  FZ,  applied  to  AD, 
is  equal  to  the  product  of  R  and  the  sum  of  the  squares 


*  This  is  the  converse  of  Lect.  X,  §  n. 


136  BARROW'S  GEOMETRICAL  LECTURES 

of  EY,  applied  to  BD ;  and  so  on  in  similar  fashion  for  the 
other  powers. 

22.  Let  DOK  be  any  curve,  D  a  fixed  point  in  it,  and  DKE 
a  chord ;  also  let  AFE  be  a  curve  such  that,  when  any  straight 
line  DMF  is  drawn  cutting  the  curves  in  M  and  F,   DS  is 
drawn  perpendicular  to  DM,  MS  is  the  tangent  to  the  curve 
DOK,  cutting  DS  in  S,  and  R  is  any  given  straight  line,  then 
DS  :  2R  -  DM2  :  DF2.     Then  the  space  ADE  will  be  equal  to 
the  rectangle  contained  by  R  and  DK.* 

23.  The  data  and  the  construction  being  otherwise  the 
same,  let  KH  and  Ml  be  drawn  perpendicular  to  the  tangents 
KT  and   MS,  meeting  DT,  DS  in   H  and  I  respectively;  and 
let  AE  be   a   curve   such  that   DE  =  ^/(DK.DH),  and  also 
DF  =  ,y(DM  .  Dl),  and   so   on.      Then    the   space   ADE   is 
equal  to  one-fourth  of  the  square  on   DK. 

24.  If  DOK  is  any  curve,  D  a  given  point  on  it,  and  DK 
any  chord  :  also  if  DZI  is  a  curve  such  that,  when  any  point 
M  is  taken  in  the  curve  DOK,  DM  is  joined,  DS  is  drawn 
perpendicular  to  DM,  MS  is  a  tangent  to  the  curve,  DP  is 
taken  along  DK  equal  to  DM,  and  PZ  is  drawn  perpendicular 
to  DK,  then  PZ  is  equal  to  DS:  in  this  case  the  space  DZI 
is  equal  to  twice  the  space  DKOD.t 

25.  The  data  and  the  construction  being  in  other  respects 
the  same,  let  the  ordinates  PZ  now  be  supposed  to  be  equal 

*  The  analytical  equivalents  arc  : 

22.  frf.  dti  =  aR  .y>2/(r2  .  dOfdr) .  </H  =  2ft r. 

23.  frf/z.dS  =/>/2  .  (dr/dti) .  dQ  =  r2/4. 

24.  'fi*.dQ  =/r2  .  (dSfdr)  .  dr. 
|  See  note  on  page  138, 


LECTURE  XI  137 

to  the  respective  tangents  MS  ;  take  any  straight  line  xk, 
and  distances  along  it  equal  to  the  arcs  DOK,  DOM,  DON, 
etc.,  and  draw  the  ordinates  kd,  md,  nd,  etc.,  equal  to  the 
chords  KD,  MD,  ND,  etc. ;  then  the  space  xkd  will  be  equal 
to  the  space  DKI. 

26.  Moreover  if,  other  things  remaining  the  same,  any 
straight  line  kg  is  taken,  the  rectangle  xkgh  is  completed,  and 
the  curve  DZI  is  supposed  to  be  such  that  MD  :  D8  =  kg:  PZ  ; 
then  the  rectangle  xkgh  will  be  equal  to  the  space  DKI. 

Hence,  if  the  space  DKI  is  known,  the  quantity  of  the 
curve  DOK  may  be  found. 

Should  anyone  explore  and  investigate  this  mine,  he  will 
find  very  many  things  of  this  kind.  Let  him  do  so  who 

must,  or  if  it  pleases  him. 

• 

Perhaps  at  some  time  or  other  the  following  theorem, 
too,  deduced  from  what  has  gone  before,  will  be  of  service  ; 
it  has  been  so  to  me  repeatedly. 

27.  Let  VEH  be  any  curve,  whose  axis  is  VD  and  base 
DH,  and  let  any  straight  line  ET  touch  it;  draw  EA  parallel 
to  HD.     Also  let  GZZ  be  another  curve  such  that,  when  any 
straight  line  EZ  is  drawn  from   E  parallel  to  VD,  cutting  the 
base  HD  in  I  and  the  curve  GZZ  in  Z,  and  a  straight  line  of 
given  length  R  is  taken,  then  at  all  times  DA2 :  R2=  DT  :  IZ. 
Then  DA  :  AE  =  R2 :  space  DIZG  ;  (or,  if  DA  :  R  is  made  equal 
to   R:  DP,  and  PQ  is  drawn  parallel  to   DH,  then  the  rect- 
angle DPQI  is  equal  to  the  space  DGZI).     [Fig.  131,  p.  139.] 

The  following  theorem  is  also  added  for  future  use. 

28.  Let  AMB  be  any  curve  whose  axis  is  AD ;  also  let  the 


138  BARROWS  GEOMETRICAL  LECTURES 

line  KZL  be  such  that,  if  any  point  M  is  taken  in  AMB,  and 
from  it  are  drawn  a  straight  line  MP  perpendicular  to  AB 
cutting  AD  in  P,  and  a  straight  line  MG  perpendicular  to  AD 
cutting  the  curve  KZL  in  Z,  at  all  times  GM  :  PM  is  equal  to 
arc  AM : GZ;  then  the  space  ADKL  will  be  equal  to  half  the 
square  on  the  arc  AM. 

These  theorems,  I  say,  may  be  obtained  from  what  has 
gone  before  without  much  difficulty ;  indeed,  it  is  sufficient 
to  mention  them ;  and,  in  fact,  I  intend  here  to  stop  for 
a  while. 

NOTE 

The  theorems  of  §§  24-28  deserve  a  little  special  notice. 
The  first  of  these  was  probably  devised  by  Barrow  for  the 
quadrature  of  the  Spiral  of  Archimedes ;  it  included,  as  was 
usual  with  him,  "innumerable  spirals  of  other  kinds,"  thus 
representing  both,  as  Barrow  would  consider  it,  an  improve- 
ment and  a  generalization  of  Wallis'  theorems  on  this  spiral 
in  the  "Arithmetic  of  Infinites." 

It  is  readily  seen  that  if  DZI  is  a  straight  line,  the  curve 
AOK  is  the  first  branch  or  turn  of  the  Logarithmic  or 
Equiangular  Spiral;  if  DZI  is  a  parabola,  the  curve  DOK  is 
the  Circular  Spiral  or  Spiral  of  Archimedes;  and  if  the 
curve  is  any  paraboliform,  the  curve  DOK  is  a  spiral  whose 
equation  may  be  rm  =  kBn.  In  short,  Barrow  has  given  a 
general  theorem  to  find  the  polar  area  of  any  curve  whose 
equation  is  0  =  f/(^)/^2.  dr,  for  all  cases  in  which  he  can 
find  the  area  under  the  curve  y  =  f(x). 

The  theorem  of  §  26  is  indeed  remarkable,  in  that  it  is  a 
general  theorem  on  rectification.  It  is  stated  *  that  Wallis 
had  shown,  in  1659,  that  certain  curves  were  capable  of 
rectification,  that  William  Neil,  in  1660,  had  rectified  the 
semi-cubical  parabola,  using  Wallis'  method,  that  the  second 
curve  to  be  rectified  was  the  cycloid,  and  that  this  was 

*  Ency.  Brit.  (Times  edition),  Art.  on  Infinitesimal  Calculus. 
(Williamson).  These  dates  are  wrong,  however,  according  to  other 
authorities,  such  as  Rouse  Ball. 


LECTURE   XI 


139 


effected  by  Sir  C.  Wren  in  1673.  Barrow's  general  theorem 
includes  as  a  special  case,  when  the  line  DZI  is  a  straight 
line,  whose  equation  isy  =  ^2  .  x,  the  curve  DOK  with  the 
relation  ds\dr  =  Ji-r,  that  is  the  triangle  DMS  is  always 
a  right-angled  isosceles  triangle,  and  therefore  the  curve  is 
the  Logarithmic  or  Equiangular  Spiral,  which  may  thus  be 
considered  to  be  the  real  second  curve  that  was  rectified. 
Even  if  not  so,  we  shall  find  later  that  Barrow  has  anticipated 
Wren  in  rectifying  the  cycloid,  as  a  particular  case  of 
another  general  theorem ;  and  in  this  case,  he  distinctly 
remarks  on  the  fact  that  he  has  done  so.  In  general, 
Barrow's  theorem  rectifies  any  curve  whose  equation  is 
0  =  f  J(R2-rt)/r2.dr,  where  R  =/<>),  so  long  as  he  can 

find  the  area  under  the  curve  y  =f  (x). 

The  theorem  of  §  27  is  even  more  remarkable,  not  only 
for  the  value  of  its  equivalent,  which  is  the  differentiation 
of  a  quotient,  but  also  because  it  is  a  noteworthy  example 
of  what  I  call  Barrow's  contributory  negligence ;  for 
although  he  recognizes  its  value,  and  indeed  states  that 
it  has  been  of  service  to  him  "repeatedly"  (and  no  wonder), 
yet  he  thinks  that  "  it  is  enough  to  mention  it,"  and 
omits  the  proof,  which  "may  be  obtained  from  what 
has  gone  before  without  much  difficulty."  Even  the  figure 
he  gives  is  the  worst  possible  to  show  the  connection,  as  it 
involves  the  consideration  that  the  gradient  is  negative 
when  the  angle  of  slope  is  obtuse.  Of  the  figures  below, 
the  one  on  the  right-hand  side  is  that  given  by  Barrow ; 


W     / 


the  proof,  which  Barrow   omits,  may  be  given  as  follows, 
reference  being  made  to  the^figure  on  the  left-hand  side. 


140  BARROW'S  GEOMETRICAL  LECTURES 


Let  the  curve  VXY  be  such  that,  if  EA  produced  meets  it 
in  Y,  then  always  EA  :  AD  =  AY  :  R.  Divide  the  arc  EV  into 
an  infinite  number  of  parts  at  F,  L,  .  .  .  ,  and  draw  FBX, 
LCX,  .  .  . ,  parallel  to  HD,  meeting  YD  in  B,  C,  .  .  .  ,  and  the 
curve  VXY  in  the  points  X  ;  also  draw  FJZ,  •  .  .  ,  parallel  to 
VD,  meeting  HD  in  J,  .  .  .  ,  and  the  curve  GZZ  in  the 
points  Z. 


Then  AY  .  AD.  BD  =  R  .  EA  .  BD  =  R.  (EA  .  AD  -  EA  .  AB), 
and  BX  .  AD  .  BD  =  R  .  FB  .  AD  =  R  .  (EA  .  AD  -  IJ  .  AD)  ; 
hence,  if  XW  is  drawn  parallel  to  VD,  cutting  AY  in  W,  then 

WY  .  AD2  =  WY  .  AD  .  BD  =  R  .  (IJ  .  AD  -  EA  .  AB). 
But        EA:AT=IJ:AB,     or     EA  .  AB  =  IJ  .  AT  ; 
WY  .  AD2  -  R  .  (IJ  .  AD  -  IJ  .  AT)  =  R  .  IJ  .  DT. 
Now  DA2  :  R2  =  DT  :  IZ  =  I  J  .  DT  :  IJ  .  IZ  ; 

R2:IJ.IZ  =  AD2:IJ.DT  -  R:WY. 

Hence,  since  the  sum  of  the  rectangles  IJ  .  IZ  only  differs 
in  the  least  degree  from  the  space  DGZI,  and  the  sum  of 
the  lengths  WY  is  AY  ;  it  follows  immediately  that 

R2:  space  DGZI  =  R  :  AY  =  DA:AE. 

Now  if  DT  and  DH  are  taken  as  the  co-ordinate  axes,  then 
WY  is  the  differential  of  AY  or  Ry/x,  and  DT  =  x  -y  .  dxjdy  ; 
therefore  the  analytical  equivalent  of  WY  .  AD2  =  R.  DT.  IJ 
is  R  .  d(y\x)  .  x*  -  R  .  (x  -y  .  dxjdy)  .  dy,  or  d(yjx)  -  (x  .  dy 


Barrow  states  it  as  a  theorem  in  integration;  but,  if  I 
have  correctly  suggested  his  method  of  proof,  he  obtains 
his  theorem  by  the  differentiation  of  yjx  (see  pages  94,  1  12). 


APPENDIX 

i.  When  many  years  ago  I  examined  the  Cyclometrica  of 
that  illustrious  man,  Christianus  Hugenius,*  and  studied  it 
closely,  I  observed  that  two  methods  of  attack  were  more 
especially  used  by  him.  In  one  of  these,  he  showed  that 
the  segment  of  a  circle  was  a  mean  between  two  parabolic 
segments,  one  inscribed  and  the  other  circumscribed,  and 
in  this  way  he  found  limits  to  the  magnitude  of  the  former. 
In  the  other,  he  showed  that  the  centre  of  gravity  of  a  cir- 
cular segment  was  situated  between  the  centres  of  gravity 
of  a  parabolic  segment  and  a  parallelogram  of  equal  altitude, 
and  hence  found  limits  for  this  point.  It  occurred  to  me 
that  in  place  of  the  parabola  in  the  first  method,  and  of 
the  parallelogram  in  the  second,  some  paraboliform  curve 
circumscribed  to  the  circular  segment  could  be  substituted, 
so  that  the  matter  might  be  considered  somewhat  more 
closely.  On  examining  it,  I  soon  found  that  this  was 


*  The  work  of  Christiaan  Huygens  (1629-1695),  the  great  Dutch  mathe- 
matician, astronomer,  mechanician,  and  physicist,  that  is  referred  to  may 
be  the  essay  Exetasis  quadratures  circuit  (Leyden,  1651),  but  more 
probably  is  the  complete  treatise  De  circuit  magnitudine  inventa,  that 
was  published  three  years  later.  Putting  the  date  of  Barrow's  study  of 
Huygens'  work  at  not  later  than  1656  (note  the  words  in  the  first  line  above 
that  I  have  set  in  \tn\\cs— many  years  ago, — and  remembering  that  this  was 
printed  in  1670),  it  follows  from  Barrow's  mention  of  the  paraboliform 
curve  as  something  well  known  to  him,  and  from  a  remark  that  the  proofs 
of  the  theorems  of  §  2  "  may  be  deduced  in  various  ways  from  what  has 
already  been  shown,  without  much  difficulty,"  that  Barrow  was  in 
possession  of  his  knowledge  of  the  properties  of  his  beloved  paraboliforms 
even  before  this  date.  Is  it  not  therefore  probable,  nay  almost  certain, 
that  Barrow,  in  1655  at  the  very  latest,  had  knowledge  of  his  theorem 
equivalent  to  the  differentiation  of  a  fractional  power  f 


142  BARROW'S  GEOMETRICAL  LECTURES 


correct ;  moreover,  I  easily  found  that  like  methods  could 
be  used  for  the  magnitude  of  a  hyperbolic  segment.  As 
the  proofs  for  these  theorems — better  perhaps  than  others 
that  might  be  invented — are  short,  and  clear  (because  they 
follow  from  or  depend  on  what  has  been  shown  above), 
I  thought  good  to  set  them  forth  in  this  place.  I  think, 
too,  that  they  are  in  other  respects  not  without  interest. 

2.  Let  us  assume  the  following  as  known  theorems ;  of 
which  the  proofs  may  be  deduced  in  various  ways  from, 
what  has  already  been  shown,  without  much  difficulty. 

If  BAE  is  a  paraboliform  curve,  whose 
axis  is  AD  and  base  or  ordinate  is  BDE, 
BT  a  tangent  to  it,  and  K  the  centre 
of  gravity  ;  then,  if  its  exponent  is  n\m, 
we  have* 

Area  of  BAE  =  m/(n  +  m)  of  AD .  BE, 

TD  =  m/n  of  AD, 
and     KD  =  m/(n  +  2m)  of  AD. 


Fig.  133- 


*  The  definition  of  Lect.  VII,  §  12,  uses  N/M  ;  the  value  of  TD/AD  is 
found  in  Lect.  IX,  §  4 ;  where  also  the  definition  of  the  paraboliforms  is 
given. 

Now  it  is  clear  from  the  adjoining  figure  that 
if  AHLE  is  a  paraboliform,  whose  exponent  is 
rls(  =  i/a  say),  then  LK/HK  =  a .  LM/AM  ;  and 
conversely. 

Let  AIFB  be  a  curve  such  that 

FM/R  =  LK/HK  =  a  .  LM/AM  ; 
then,  by  Lect.  XI,  §  19,  area  AFBD  =  R  .  DE. 
But,  in  this  case,  we  have 

IG  :  FM  =  LM/AM  -  HN/AN  :  LM/AM 

-  AM . LK-LM .  HK : LM  . AN 
=  (a-i).  LM  .  HK  :  LM.  AN  ; 

FG/GI  =  i/(a-i)of  AM/FM! 

Hence  AIFB   is  a  paraboliform,  whose  vertex  is  A,  axis  AD,  exponent 

a-  i. 


M 


B 


LECTURE  XI— APPENDIX  143 

3.  Let  AEB,  AFB  be  any  two  curves,  having  the  axis  AD 
and  the  ordinate  BD  common,  so  related  that,  if  any  straight 
line  EFG  is  drawn  parallel  to  BD,  cutting  the  given  lines  in 
the  points  E,  F,  G,  and  E8,  FT  touch  the  curves  AEB,  AFB 
respectively,  TG  is  always  greater  than  SG  ;  then  I  say  that 
no  part  of  the  curve  AFB  can  fall  within  the  curve  AEB. 

4.  Let  BAE  be  any  curve,  of  which  AD  is  the  axis,  and  le: 
the  base  ADE  be  an  ordinate  to  it;  also  let  the  point  H  be 
the  centre  of  gravity  of  the  segment  BAE,  and  RS  a  straight 
line  through  it  parallel  to  BE.     Further  let  another  curve 
(or  any  line  you  please)  MRASN  pass  through  the  points 
R,  8,  and  have  the  same  axis  ADj  let  it  cut  the  first  curve 
BAE  in  such  a  manner  that  the  upper  part   RKAPS  falls 
within  the  curve  BAE,  but  the  lower  remaining  parts,  RM, 
SN,  fall  outside  it.     Then  the  centre  of  gravity  of  the  seg- 
ment MRASN  will  be  below  the  point  H,  that  is,  towards  the 
base  MN. 

5.  Let  the  two  straight  lines  BT,  ES  touch  the  circle  AEB, 
whose  centre  is  C,  and  meet  the  diameter  CA  in  the  points 
T  and  8  ;  also  let  the  straight  lines  BD,  EP  be  perpendicular 
to  CA.     Then,  if  AD  >  AP,     TD  :  AD  >  SP  :  AP. 

Conversely,  if  AIFB  has  an  exponent  n}m(  =  a  -  i),  the  integral  curve  is  a 
paraboiiform,  exponent  i/a  or  m/(n  +  m). 

Hence,  since  DB/R  =  <z .DE/AD,  area  Al FED  =  R.  DE  =  m/(n  +  m)  of  AD.DB. 

Similarly,  area  ALED  =  AD.  DE-(n  +  m)l(n  +  zm)  of  AD  .  DE, 

=  m\(ii  +  2/;i)  of  AD  .  DE  ; 
R  .  a  .  area  ALED  :  AD  .  area  AIBD  =  n  +  m  :  n  +  2m. 

Now,   since  FM/R  =  a  .  LM/AM,     .  •.  FM  .  AM  .  MN  =  R  .  a  .  LM  .  HK  ; 
hence,  summing,  AK  .  area  AFBD  =  R  .  a  .  area  ALED  ; 
therefore  AK  :  AD  =  n  +  m  :  n-\-2.m,     or     KD  =  f?tf(tt+am)  of  AD. 

In  a  similar  way  the  centre  of  inertia  could  be  found. 

The  proof  could  have  been  deduced  from  the  note  on  §  2  of  Lect.  XI 
or  by  drawing  a  subsidiary  curve  as  in  the  note  to  §  27  of  Lect.  XI. 


144  BARROW'S  GEOMETRICAL  LECTURES 

6.  Let  the  two  straight  lines  BT,  ES  now  touch  a  hyper- 
bola AEB,  whose  centre  is  C;  and  let  other  things  be  the 
same  as  in  the  theorem  just  before ;  then  TD  :  AD  >  SP  :  A  P. 

7.  Let  the  axis  AD  and  the  base  BD  be  common  to  the 
circle  AEB  whose  centre  is  C,  and  the  paraboliform  AFB  ; 
also  let  the  exponent  of  the  paraboliform  be  n/m,  where 

AD  =  (m-  2ti)l(m-ri)  of  CA, 
or  m  -  n  :  m  -  2n  =  CA  :  AD. 

Moreover,  let  the  straight  line  BT  touch  the  circle ;  then  BT 
will  touch  the  paraboliform  also. 

8.  It  should  be  noted  in  this  connection  that,  conversely, 
if  the  ratio  of  AD  to  CA  is  given,  the  paraboliform  which 
touches  the  circle  AEB  at  B  is  thereby  determined. 

For  instance,  if  AD/CA  =  *//,  then  (t-s)l(2t-s)  will  be 
the  exponent  of  the  required  paraboliform. 

9.  With  the  same  hypothesis  as  in  §  7,  the  paraboliform 
AFB  will  fall  altogether  outside  the  circle  AEB. 

10.  Again  with  the  same  hypothesis,  if  with  a  base  GE 
(any  parallel  to  BD)  and  axis  AD  another  paraboliform  is 
supposed  to  be  drawn,  of  the  same  kind  as  AFB  (or  having 
the  same  exponent  njm) ;  then  this  curve  also,  for  the  part 
AE  above  GE,  will  fall  altogether  outside  the  circle. 

11.  Also  it  may  be  shown  that  the  said  paraboliform  (of 
like  kind  to  AFB  and  constructed  on  the  base  GE),  when 
produced  below  GE  to  DB,  will  fall  altogether  within  the 
circle  as  regards  this  part. 

12.  Further,  let  AD  be  the  axis  and  DB  the  base  common 


LECTURE  XI— APPENDIX  145 

to  the  hyperbola  AEB  whose  centre  is  C,  and  the  para- 
boliform  AFB,  whose  exponent  is  n/m;  also  let  AD  = 
(2H-m)/(m-n)  of  CA ;  and  let  BT  touch  the  hyperbola. 
Then  BT  touches  the  paraboliform  also. 

13.  Hence   again,  if  the   ratio    of  AD  to    CA   is   given, 
the  paraboliform  touching   the  hyperbola   at   the   point  B 
is    thereby    determined.      For    instance,    if    AD/CA  =  s/f, 
n\m  =  (t  +  s)/(2t  +  s). 

14.  With  the  same  hypothesis  as  in  §  12,  the  paraboliform 
AFB  will  lie  altogether  within  the  hyperbola  AEB. 

15.  Also,  with    the  same    hypothesis,  if  you   imagine  a 
paraboliform  of  the  same  kind  to  be  constructed  with  the 
base  GE  and  axis  AG  ;  it  will  fall  within  the  hyperbola  on 
the  upper  side  of  GE. 

1 6.  Moreover,  if  this  second  like  paraboliform,  constructed 
on  the  base  EG,  is  supposed  to  be  produced  to  DB ;  then 
the  part  of  it  intercepted  by  EG  and  BD  will  fall  altogether 
outside  the  hyperbola. 

17.  Let  the  circle  AEB  and  the  parabola  AFB  have  a 
common  axis  AD  and  base  BD;  then  the  parabola  will  fall 
within  the  circle  on  the  side   above  BD,  and  without  the 
circle  below  BD. 

If  an  ellipse  is  substituted  for  the  circle,  the  same  result 
holds  and  is  proved  in  like  manner. 

1 8.  Let  the  hyperbola  whose  axis  is  AZ  and  parameter 
AH,  and  the  parabola  AFB  have  the  same  axis  AD  and  base 
BD;    then    the    parabola   will    fall    altogether   outside    the 

10 


146  BARROW'S  GEOMETRICAL  LECTURES 

hyperbola     above     BD,     but    within     it     when    produced 
below  BD. 

19.  From  what  has  been  said,  the  following  rules  for  the 
mensuration  of  the  circle  may  be  obtained. 

Let  BAE  be  a  part  of  a  circle,  of  which  the  axis  is 
AD,  and  the  base  BE;  let  C  be  the  centre  of  the  circle,  and 
EH  equal  to  the  right  sine  of  the  arc  BAE ;  also  let 
AD  :  CA  =  s  :  t. 

Then     (i)  (2/-  s)/(^t  -  25)  of  AD  .  BE  >  segment  BAE  ; 

(2)  EH  +  (4/  -  2s)/(st  -  2s)  of  BH  >  arc  BAE  ; 

(3)  2/3  of  AD  .  BE  <  segment  BAE  ; 

(4)  EH +4/3  of  BH  <arc  BAE. 

20.  Similarly,  the  following  rules  for  the  mensuration  of 
the  hyperbola  may  be  deduced. 

Let  ADB  be  a  segment  of  a  hyperbola  [Barrow's  figure  is 
really  half  a  segment],  whose  centre  is  C,  axis  AD,  and  base 
DB;  and  let  AD  :  CA  =  sit. 

Then  (i)  (2t  +  s)/($t+2s)  of  AD  .  DB  <  segment  ADB; 
(2)  2/3  of  AD  .  DB  >  segment  ADB. 

NOTE 

The  results  of  §§  19,  20,  for  which  Barrow  omits  any  hint 
as  to  proof,  are  thus  obtained. 

§  19  (i)  A  paraboliform  whose  exponent  is  (t  —  s)J(2t-s) 
can  be  drawn,,  touching  the  circle  BAE  at  B,  A,  and  E,  and 
lying  completely  outside  it ;  the  area  of  it  cut  off  by  the 
chord  BE  is,  by  §  n,  equal  to  (2t-  2s)/(^t-2s)  of  AD  .  BE. 
(3)  A  parabola  is  a  paraboliform  whose  exponent  is  1/2,  and 
the  area  of  the  segment  is  2/3  of  AD  .  BE.  (2)  and  (4) 
follow  from  (i)  and  (3)  by  using  obvious  relations  for  the 
circle,  and  are  not  obtained  independently.  This  explains 


LECTURE  XI— APPENDIX  14; 

why  there  are  only  two  formulae  given  for  the  hyperbola,  and 
these  are  formulas  for  the  segment;  for  there  are  no 
corresponding  simple  relations  for  the  hyperbola  that 
connect  the  sector  or  segment  with  the  arc. 

§  20.  In  a  similar  way,  the  two  limits  for  the  hyperbolic 
segment  are  obtained  from  a  paraboliform  whose  exponent 
is  (t  +  s)j(2t  +  s),  and  a  parabola. 

The  formulae  of  (i)  and  (2)  for  the  circle  reduce  to  the 
trigonometrical  equivalent  a  <  sin  a .  (2  +  cos  a)/(i  +  2  cos  a) 
in  which  the  error  is  approximately  a5/45  ;  the  formulae 
of  (3)  and  (4)  reduce  to  the  much  less  exact  equivalent 
a  >  sin  a.  (2 -\-cos  a)/3,  where  a  is  the  half-angle.  Thus 
Barrow's  formula  is  a  slightly  more  exact  approximation 
than  that  of  Snellius,  namely,  3  sin  2a/2(z+<:0s  20),  where 
the  error  is  approximately  4^/45,  and  is  in  defect;  Barrow, 
in  §  29,  obtains  Snellius'  formula  in  the  more  approxi- 
mate form  $  sin  a.l(2+ cos  a.).  Hence  Barrow's  formula  and 
Snellius'  formulae  give  together  good  upper  and  lower 
limits  to  the  value  of  the  circular  measure  of  an  angle. 
The  equivalent  to  the  first  formula  for  the  hyperbola  is 
sin~l  (tan  a)  >  3  tan  a/(i  +2  cos  a) ;  the  error  being  again  of 
the  order  a5. 

21.  Further,  let  BAE  be  the  segment  oif  a  circle  whose 
centre  is  C,  axis  AD,  centre  of  gravity  K  ;  also  let  AD  :  CA  = 
s:f,  and  HD  :  AD=2/- s  :  5/-  2s ;  then  HD  will  be  greater 
than  KD.* 

22.  Let   the    point    L  be    the   centre   of  gravity  of   the 
parabola  (such  as  was  discussed  in  §  18);  then  L  will  be 
below  K  ;  i.e.  KD  is  greater  than  two-fifths  of  AD.* 

23.  Let  BAE  be  a  segment  of  a  hyperbola  whose  centre 
is  C,  axis  AD,   base  BE,  and  centre  of  gravity  K;  also  let 
AD:  CA  =  J:/,  and  HD  :  AD  =  zt  +  s:  5/+2j;  then  HD  is  less 
than  KD.* 

*  See  note  at  foot  oi  next  page. 


H8  BARROW'S  GEOMETRICAL  LECTURES 

24.  The   centre    of  gravity  of  the    parabola,   L  say,  lies 
above  K ;  i.e.  KD  is  less  than  two-fifths  of  AD.* 

25.  Lest  the  present  method  of   research,  owing  to  the 
great   number  of  methods  of  this  kind  for  measuring  the 
circle,  may  seem  to  be  of  little  account,  we  will  add  one  or 
two  riders  (if  only  for  the  sake  of  these,  the  few  theorems 
given   would    deserve   employment) ;    from    which   indeed 
Maxima  and  Minima  of  things  of  a  kind  may  be  determined 
in  a  great  number  of  cases. 

Let  ABZ  be  a  semicircle  whose  centre  is  C  ;  also  let  ADB 
be  a  segment;  and  to  this  let  a  paraboliform  AFB  be 
adscribed,  whose  exponent  is  n/m,  where  AD :  CA  = 
m  —  2/1:  m  -  n. 

If  the  parameter  of  the  paraboliform  f  (that  is  a  straight 
line  such  that  some  power  of  it  multiplied  by  a  power  of 
the  axis  of  the  segment,  AD  say,  produces  a  power  of  the 
ordinate,  DB  say)  is  / ;  then  p  will  be  a  maximum  of  its 
kind. 

For,  if  any  straight  line  GE  is  drawn  parallel  to  DB,  and 
a  paraboliform  of  like  kind  to  AFB  is  supposed  to  be 
applied  to  GE,  of  which  the  parameter  is  called  q\  then, 
since  the  paraboliform  AFB  touches  the  circle  externally, 
GF  >  GE.  /.  GFW  >  GEm,  or  /»-»  .  AG"  >  gm~n  .  AG", 

•'•/  >?- 

It  should  be  observed  that  /""-«>  =  ZDW  .  AD""2'1  and 

*  These  limits  are  not  remarkable  for  close  approximation  unless  the 
segment  is  very  shallow.  Thus  if  the  arc  is  one-third  of  the  circumference, 
the  limits  for  the  circle  are  only  2AD/5  and  3AD/5. 

f  It  is  to  be  observed  that  Barrow  here  indicates  that  the  equation  to  the 
paraboliform  is,  in  general  y  =  axm/n. 


LECTURE  XI— APPENDIX  149 

f(m-n)  =  zQm  .  AG"1"2",  hence   ZDm  .  ADm"2n  >  ZGm  .  AGm~2n ; 
that  is  ZDm .  AD"1"2"  is  a  maximum.* 

Example  i.  Let  »  =  I,  and  w  -  3;  then/4  -  ZD3 .  AD  - 
ZD2 .  BD2,     or    /2  =  ZD  .  BD ;     and     AD  -  CA/2. 

Example  2.  Let  «  =  3,  and  m  =  10;  then/14  =  ZD10.  AD4, 
or  /  -  ZD5 .  AD2  -  ZD3 .  BD4,  and  AD  -  4CA/7. 

26.  Again,  let  AEB   be  an   equilateral  hyperbola  whose 
centre  is  C,  and  axis  ZA ;  and  to  it  let  a  paraboliform  AFB, 
whose  exponent  is   njm   and    parameter  /,    be   adscribed 
(with  a  base  DB) ;  also  suppose  that  AD  :  CA  =  2n-m\m-n\ 
then  p  will  be  a  minimum  of  its  kind. 

It  is  to  be  noted  that  /<™-">  =  ZDm .  AD2n"TO,  and  also 
(as  in  §  25)  $-{m-n}  =  ZGm.  AG2" ;  hence  ZDm .  AD""  is 
a  minimum.! 

As  in  the  preceding  I  have  touched  upon  the  mensura- 
tion of  the  circle,  what  if  I  add  incidentally  a  few  theorems 
bearing  upon  it.  which  I  have  by  me?  The  following  general 
theorem  must,  however,  be  given  as  a  preliminary. 

27.  Let  AGB  be  any  curve  whose  axis  is  AD,  and  let  the 
straight  lines  BD,  GE  be  ordinates  to  it.     Then  the  arc  AB 
will  bear  a  greater  ratio  to  the  arc  AG  than  the  straight  line 
BD  to  the  straight  line  GE. 


*  This  is  equivalent  to  the  algebraical  theorem  that,  if  x+y  =  a  constant 
then  xr  .  ys  is  a  maximum  when  xjr  =  y/s. 

f  This  is  equivalent  to  the  algebraical  theorem  that,  if  x-y  =  a  constant, 
then  x^jy8  is  a  minimum  when  x\r  =  y/s. 

The  proof  of  this  theorem  is  generally  ascribed  to  Ricci,  who  proved  it 
algebraically  in  1666,  and  used  it  to  draw  the  tangent  to  the  general 
paraboliform  ;  thus  we  see  that  Barrow's  proof  was  independent  of  Ricci, 
even  if  Barrow  had  not  discovered  it  before  Ricci ;  cf.  "  many  years  ago." 


150  BARROW'S  GEOMETRICAL  LECTURES 

28.  Let  AM  B  be  a  circle,  of  which  the  radius  is  CA,  and 
let  DBE  be  a  straight  line  perpendicular  to  CA  ;  also  let  ANE 
be  a  curve  such  that,  when  any  straight  line  PMN  is  drawn 
parallel  to  DE,  cutting  the  circle  in  M  and  the  curve  in  N, 
the  straight  line  PN  is  equal  to  the  arc  AM.     Then  the  para- 
bola described  with  axis  AD  and  base  DE  will  fall  altogether 
outside  the  curve  ANE. 

29.  From  the  preceding,  and  from  what  is  commonly 
known  about  the  dimensions  of  the  spaces  ADB,  ADE,*  the 
following  formula  may  be  easily  obtained  : 

3CA  .  DB/(2CA  +  CD)  <  arc  AB. 

Further,  if  the  arc  AB  is  supposed  to  be  one  of  30 
degrees,  and  2CA  =  113,  then  the  whole  circumference, 
calculated  by  this  formula,  will  prove  to  be  greater  than 
355  less  a  fraction  of  unity. 

30.  Hence  also,  being  given  the  arc  AB,  let  arc  AB  =  /, 
CA  =  r,  and  DB  =  e,  then  the  following  equation  may  be 
used  to  find  the  right  sine  DB  : 


or,  substituting  k  for  ^rip\(^r'L  +/2),  we  have 

kp  =  4&e  -  <?2  ;     or     2k-  J(4&  -  kp)  =  e. 

31.  Let  AMB  be  a  circle  whose  radius  is  CA,  and  let  the 
straight  line  DBE  be  perpendicular  to  CA  ;  let  also  the  curve 
ANE  be  a  part  of  the  cycloid  pertaining  to  the  circle  AMB  ; 
and  lastly  let  a  parabola  AOE  be  drawn  with  axis  AD  and 
base  DE.  Then  the  parabola  will  fall  altogether  within  the 
cycloid. 

*  See  note  at  the  end  of  this  lecture. 


LECTURE   XI— APPENDIX  151 

32.  From    the   preceding,    and    from    what   is   generally 
known  about  the  dimensions  of  circles  and  cycloids,*  the 
following  formula  may  be  obtained ; 

(2CA  .  DB  +  CD  .  DB)/(CA  +  2CD)  >  arc  AB. 

Further,  if  the  arc  AB  is  one  of  30  degrees,  and  2CA  =  1 1 3> 
it  may  be  shown  by  this  formula  that  the  whole  cir- 
cumference is  less^than  355  plus  a  fraction. 

You  see  then  that,  from  the  two  formulae  stated,  there 
results  immediately  the  proportion  of  the  diameter  to  the 
circumference  as  given  by  Metius. 

33.  Since  in   this   straying   from  the   track,   the  cycloid 
has  brought  itself  under  notice,  I   will  add  the  following 
theorem ; — I    am    not   aware   that    it   has    been    anywhere 
observed   by  those  who  have  written  so  profusely  on  the 
cycloid. 

If  the  space  ADEG  is  completed  (in  §  31),  the  space  AEG 
will  be  equal  to  the  circular  segment  ADB. 

The  proof  I  shall  leave  out,  nor  shall  I  wander  further 
from  my  subject. 

34.  Let  two  circles  AIMG,  AKNH  touch  one  another  at  A, 
and  have  a  common  diameter  AHG  ;  and  let  any  straight 
line    DNM    be    drawn    perpendicular   to    AHG.      Then    the 
segment  AIMD  will  bear  to  the  segment  AKNH  a  less  ratio 
than  the  straight  line  DM  to  the  straight  line  DN. 

35.  Let  YFZT  be  an  ellipse,  of  which  YZ  and  HT  are  the 
conjugate  axes ;  and  let  the  straight  line  DC  be  parallel  to 

*  See  note  at  the  end  of  this  lecture. 


152  BARROW'S  GEOMETRICAL  LECTURES 

the  major  axis  YZ,  and  let  the  circle  DFCV,  whose  centre  is 
K  a  point  on  the  minor  axis  FT,  pass  through  the  points 
D,  F,  C;  then  I  say  that  the  part  DOFPC  of  the  circle  will 
lie  within  the  part  DMFNC  of  the  ellipse. 

36.  Let  DEC  be  a  segment  of  a  circle  whose  centre  is  L; 
and,  any  point  F  being  taken  in  its  axis  GE,  let  DMFC  be  a 
curve   such   that,   when  any  straight   line    RMS   is   drawn 
parallel  to  GE,  R8  :  RM  =GE  :  GF;  then  DMFC  is  an  ellipse 
thus  determined  :— Find   H,  such  that  EG  :  FG  =  GL  :  GH  ; 
through  H  draw  YHZ  parallel  to  DC,  and  let  HY  equal  LE ; 
then  HY,  HF  are  the  semi- axes  of  the  ellipse. 

This  is  held  to  have  been  proved  by  Gregory  St  Vincent, 
Book  IV,  Prop.  154. 

COROLLARY. — Hence,  segment  DEC  :  segment  DMFC 
=  EG  :  FG. 

37.  Let  DEC,  DOFC  be  portions  of  two  circles  having  a 
common  chord  DC  and  axis  GFE;  then  the  greater  portion 
DEC  will  bear  to  the  portion  DOFC  a  greater  ratio  than  that 
which  the  axis  GE  bears  to  the  axis  GF. 

NOTE 

In  §  29,  Barrow  gives  no  indication  of  the  source  of  the 
"  known  dimensions,"  and  there  is  also  probably  a  misprint ; 
for  the  "spaces  ADB,  ADE,"  we  should  read  the  "spaces 
AN  ED,  AOED,"  unless  Barrow  intended  ADE  to  stand  for 
both  the  latter  spaces.  If  so,  we  have  from  §  2,  area 
AOED  =  2/3  of  AD .  DE,  and  the  area  of  AN  ED  can  thus  be 
found  by  Barrow's  methods : — 

Complete  the  rectangle  EDCF,  and  draw  QRS  parallel  and 
indefinitely  near  to  PMN;  draw  SVZ  parallel  to  AC,  cutting 


LECTURE   XI— APPENDIX 


153 


\ 

x^ 

/N  J 

/' 

s/ 

R          \ 

p 

\ 

•     Z     Y 
Fig.  151- 

PN,  CF  in  V,  Z,  and  RT  parallel  to  AC,  cutting  PN,  CF  in 
T,  Y ;  then  we  have  CP  :  CM  =  MT  :  MR  =  MT  :  NV, 

.-.  CP.NV  =  CM.  MT. 
Hence     area  AN  ED  +  CD.  DE 
=  the  sum  of  VN  .  CP+  .  .  . 
=  the  sum  of  CM.  MT  +  .  .  . 
=  CM  .(the  sum  of  MT+  .  .  .) 
= CM.BD; 

AreaANED  -  CA  .  BD  -  CD  .  DE. 
.-.  CA.  BD- CD.  DE<  area  AGED 

<  2/3  of  AD.  DE, 
.-.  3CA.BD<(3CD  +  2AD).DE 
or     (2CA  +  CD).arc  AB. 
It  should  be  observed  that  the  equivalent  of  the  expres- 
sion for  the  area  A  NED  is 

Ja  fos~l  x/a  dx  =  x  cos~l  x/a  -  a  .  v/(#2  -  #2). 

The  formula  finally  obtained  by  Barrow,  if  we  put  2<f>  for 
the  angle  subtended  at  the  centre  of  the  circle  by  the  arc 
AB,  reduces  to  2</>  >  3  sin  2<f>/(2  -\-cos  2<f>),  which  is  the 
formula  of  Snellius ;  this,  as  I  have  already  noted,  has  an 
error  of  the  order  <£5 ;  a  handier  result  is  obtained  by 
taking  a  =  2</>,  when  it  becomes  a  >  3  sin  a/(2  +  cos  a). 

For  §  32,  since  MN  (of  this  theorem)  =  arc  AM  =  PN  (in 
fig.   151);  hence 
area  of  cycloid  =  area  of  AM  BD  +  area  of  A  NED  (fig.  151) 

=  (CA  .  arc  AB  -  CD  .  BD)/2  +  CA  .  BD  -  CD  .  arc  AB. 
The   remark    made    by    Barrow    in  §  33  indicates   with 
almost  certainty  that  the  above  was   his   method  for  the 
cycloid. 

Now,  since  the  area  of  the  cycloid  is  less  than  the  area 
of  the  corresponding  parabola,  which  is  2/3  of  AD  .  DE  or 
2/3  of  AD  .  (DB  +  arc  AB) ;  hence  we  obtain 

arc  AB  <  (2CA  .  DB  +  CD  .  DB)/(CA  +  2CD). 

This  is  equivalent  to  a  <  sin  a(2  +  cos  a)/(i  +2  cos  a),  a 
limit  obtained  before  in  §  19.  Thus  Barrow  has  here  two 
very  close  limits,  one  in  excess  and  the  other  in  defect, 
each  having  an  error  of  the  order  a5. 


154  BARROW'S  GEOMETRICAL  LECTURES 

The  results 'Obtained,  by  the  use  of  these  approximate 
formulae,  with  the  convenient  angle  of  30  degrees  are  in 
fact  355'6  and  354*8.  The  formula  obtained  by  "Adrian, 
the  son  of  Anthony,  a  native  of  Metz  (1527),  and  father  of 
the  better  known  Adrian  Metius  of  Alkmaar  "  is  one  of  the 
most  remarkable  "  lucky  shots  "  in  mathematics.  By  con- 
sidering polygons  of  96  sides,  Metius  obtained  the  limits 
3rW  and  ST-TOJ  and  then  added  numerators  and  de- 
nominators to  obtain  his  result  3-2\2e  or  3T1T%  !  !  ! 

Barrow  seems  to  be  content,  as  usual,  with  giving  the 
geometrical  proof  of  the  formula  obtained  by  Metius ;  which 
must  have  appeared  atrocious  to  him  as  regards  the  method 
by  means  of  which  the  final  result  was  obtained  from  the 
two  limits.  rjf  only  Barrow  had  not  had  such  a  distaste  for 
long  calculationSjlsuch  as  that  by  which  Briggs  found  the 
logarithm  of  2  (he  extracted  the  square  root  of  1*024  forty- 
seven  times  successively  and  worked  with  over  thirty-five 
places  of  decimals),  it  would  seem  to  be  impossible  that 
Barrow  should  not  have  had  his  name  mentioned  with  that 
of  Vieta  and  Van  Ceulen  and  others  as  one  of  the  great 
computers  of  TrJ  For  he  here  gives  both  an  upper  and  a 
lower  limit,  and  therefore  he  is  only  barred  by  the  size  of 
the  angle  for  which  he  can  determine  the  chord.  Now,  he 
would  certainly  know  the  work  of  Vieta;  and  this  would 
suggest  to  him  that  a  suitable  angle  for  his  formulae  would 
be  7T/2n,  where  n  was  taken  sufficiently  large.  For  Vieta's 
work  would  at  once  Ijfcad  him  to  the  formulae 

2  COS   7T/2'1   =   V[>+  x/{2+x/(2+ )}]> 

2  sin  7T/2"  =  Jfp  -  ^{2  +  ^(2  + )}], 

where  there  are  n  -  i  root  extractions  in  each  case. 

If,  then,  he  took  n  to  be  48,  his  angle  would  be  less  than 
i/246,  and  the  error  in  his  values  would  be  less  than  i/2<234; 
this  is  about  io~45;  hence  Barrow  has  practically  in  his 
hands  the  calculation  of  it  to  as  many  decimal  places  as  the 
number  of  square  root  extractions  he  has  the  patience  to 
perform  and  the  number  of  decimal  places  that  he  is  willing 
to  usej 


LECTURE    XII 


General  theorems  on  Rectification, 

GENERAL  FOREWORD. — We  will  now  proceed  with  the 
matter  in  hand ;  and  in  order  that  we  may  as  far  as 
possible  save  time  and  words,  it  is  to  be  observed  every- 
where in  what  now  follows  that  AB  is  some  curved  line, 
such  as  we  shall  draw,  of  which  the  axis  is  AD ;  to  this 

Q   C  A         K 


[VI 

N 
/ 

^- 

"^^\   H 

•^~ 

R 

\ 

FD       Y\\ 

B 

t: 

S    \ 

P                     IL 

/" 

V 

0 

/ 

X 

T 

Fig.  156. 

Fig.  157. 


axis  all  the  straight  lines  BD,  CA,  MF,  NG  are  applied 
perpendicular;  the  arc  MN  is  indefinitely  small;  the 
straight  line  a/3  =  arc  AB,  the  straight  line  a/*  =  arc  AM,  and 
/xv  =  arc  MN  ;  also  lines  applied  to  a/?  are  perpendicular 
to  it.  On  this  understanding, 

i.  Let   MP  be  perpendicular  to  the  curve  AB,  and  the 


156  BARROWS  GEOMETRICAL  LECTURES 


lines  KZL,  a<£S  such  that  FZ  =  MP  and  n<j>  =  MF.    Then  the 
spaces  a/38,  ADLK  are  equal.* 

2.  Hence,  if  the  curve  AMB  is  rotated  about  the  axis  AD, 
the  ratio  of  the  surface  produced  to  the  space  ADLK  is  that 
of  a  circumference  of  a  circle  to  its  diameter;  whence,  if 
the  space  ADLK  is  known,  the  said  surface  is  known. 

Some  time  ago  we  assigned  the  reason  why  this  was  so. 

Q   C 


x 

A] 

/ 

^- 

--^\     H 

G                \\J7 

\ 

R 

r 

Y\\ 

/ 

\ 

D 

\\ 

B 

L 

S    \ 

P 

1  L 

^ 

V 

O 

/ 

X 

T 

Fig.  156. 


Fig.  157. 


3.  Hence  the  surfaces  of  the  sphere,  both  the  spheroids, 
and  the  conoids  receive  measurement.!     For,  if  AD  is  the 
axis  of  the  conic   section   from  which  these  figures  arise, 
there  always  exists  some  one  line  of  the  conies,  KZL,  that 
can  be  found  without  much  difficulty.     I  merely  state  this, 
for  it  is  now  considered  as  common  knowledge. 

4.  With   the  same  hypothesis,  let  AYI   be  a  curve  such 
that  the  ordinate  FY  is  a  mean  proportional  between  the 
corresponding    FM,    FZ.      Then    the  solid  formed   by   the 
rotation  of  the  space  aS/3  about  the  axis  a/3  will  be  equal 


*  The  equivalent  is  yds  =  y .  (daldx)dx. 

f  For  the  circle,  the  figure  ADLK  is  a  rectangle  ;  and  the  area  of  a  zone 
is  immediately  cleducible  ;  and  so  on. 


LECTURE   XII  157 

to  the  solid  formed  by  the  rotation  of  the  space  ADI  about 
the  axis  AD. 

5.  By  similar  reasoning,  it  may  be  deduced  that,  if  FY 
is  supposed  to  be  a  bimedian  between  FM  and  FZ,  the  sum 
of  the  cubes  of  the  applied   lines,   such  as  //.<£,  from  the 
curve  a<f>8,  to  the  straight  line  a/3  is  equal  to  the  sum  of 
the  cubes  on  the  lines  applied  to  the  straight  line  AD  from 
the   curve    AYI.      Similarly,    the    theorem    holds  for  other 
powers. 

6.  Further,  with  the  same  hypothesis,  let  the  curve  VXO 
be  such  that  EX  =  MP;  and  let 'the  curve  -n^  be  such  that 
fig  =  PF.     Then  the  space  airf/l  =  the  space  DVOB. 

7.  Observe   also    that,    if   the   curve   AB   is  a  parabola, 
whose  axis  is   AD  and  parameter  R ;  then  the  curve  VXO 
will  be  a  hyperbola,  whose  centre  is  D,  semi-axis  DV,  and 
the   parameter  of  this   axis   equal   to   R.     Also   the   space 
a/fy7r  will   be  a  rectangle.     Hence  it  follows  that,  being 
given  the  hyperbolic  space  DVOB,  is  to  be  given  the  curve 
A  MB,  and  vice-versa.     All  this  is  remarked  incidentally.* 

8.  It   should  also   be    possible  to  observe   that  all  the 
squares  on  the  lines  applied  to  the  straight  line  a/>,  taken 
together,  from  the  curve  TT^,  are  equal  to  all  the  rectangles 
such  as  PF.  EX,  applied  to  the  line  DB  (or  calculated);  the 
cubes  on  /x,£  are  equal  to  the  sum  of  PF2 .  EX,  etc. ;  and 
so  on. 

*  Yet  it  has  an  important  significance  ;  for  it  is  the  first  indication  that 
Barrow  is  seeking  the  connection  between  the  problem  of  the  rectification 
of  the  parabola  and  that  of  the  quadrature  of  the  hyperbola.  He  is  not 
quite  satisfied  with  this  result,  but  finally  succeeds  in  §  20,  Ex.  3. 


158  BARROW'S  GEOMETRICAL  LECTURES 

9.  Also  it  may  be  noted  that,  PMQ  being  produced,  if  FZ 
is  supposed  to  be  equal  to  PQ,  and  /*</>  to  AQ  ;  then  the 
space  afiS  is  equal  to  the  space  ADLK. 

10.  Further,  let  the  straight  line  MT  touch  the  curve  AB, 
and  let  the  curves   DXO,  a<£S  be  such  that  hX  =  MT  and 
/*0  =  MF.    Then  the  space  a/38  is  equal  to  the  space  DXOB. 


Fig.  158.  Fig.  159. 

1 1 .  Hence  again,  the  surface  of  the  solid  formed  by  the 
rotation  of  the  space  ABD  about  the  axis  AD  bears  to  the 
space  A  DOB  the  ratio  of  the  circumference  of  a  circle  to 
its  radius ;  therefore,  if  one  is  known,  the  other  becomes 
known  at  the  same  time. 

Hence  again  one  may  measure  the  surfaces  of  spheroids 
and  conoids. 

12.  If  the  line  DYI  is  such  that  EY2  =  EX.  MP;  then  the 
solid  formed  by  the  rotation  of  the  space  af3S  about  the 
axis  a/3  is  equal  to  the  solid  formed  by  the  rotation  of  the 
space  DBI  about  the  axis  IB. 

13.  By  similar  reasoning,  one  may  compare  the  sums  of 
the  cubes  and  other  powers  of  the  ordinates  with   spaces 
computed  to  the  straight  line  DB. 


LECTURE   XII 


159 


14.  Moreover,    let   the   lines    AZK,    afy    be    such    that 
FZ  =  MT  and  ju,£  =  TF;  then  the  space  a/fy  will  be  equal 
to  ADK. 

15.  Also  the  sum  of  the  squares  on   the  applied   lines 
/u£  will  be  equal  to  the  sum  of  the  rectangles  TF . FZ ;  the 
sum  of  the  cubes  on  /x,£  to  the  sum  of  TQ .  FZ,   .  .  .  (con- 
sidering them  to  be  computed  to  the  straight  line  AD) ;  and 
so  on  for  the  other  powers. 

1 6.  Again  let  the  straight  line  QMP  be  perpendicular  to 
the  curve  AMB;  and  let  /3S=BD;  complete  the  rectangle 
a{38£;  then  let  the  curve  KZL  be  such  that  FZ  =  QP.    Then 
the  rectangle  a/38£  is  equal  to  the  space  ADLK. 

Therefore,  if  the  space  AKLD  is  known,  the  quantity  of 
the  curve  AMB  is  also  known. 


0       L 

I 

?           B/v 

J       /              ° 

i 

Q   /» 

7  /z 

RlAr 

z 

V 

X            K                   A  T           C            ^ 

Fig.  1  60. 

o 

1 
Fig.  161. 

17.  Also,  let  the  straight  line  TMY  be  supposed  to  touch 
the  curve  AMB,  and  let  /3y  be  made  equal  to  BC,  and  the 
rectangle  a/?y^  be  completed ;  let  then  the  curve  OXX  be 
such  that  FX  =  TY.  Then  the  space  ADOXX  .  .  .—in- 
definitely continued — will  be  equal  to  the  rectangle  af3yif/. 

Hence,  again,  if  the  space  ADOXX  .  .  .  has  been  ascer- 
tained, then  the  curve  AMB  becomes  known. 


160  BARROW'S  GEOMETRICAL  LECTURES 


1 8.  Moreover,  if  any  determinate  length   R  is  taken,  and 
(38  is  taken  equal  to  R ;  and  if  the  curve  OXX  is  such  that 
MF:MP  =  R  to  FX;  then  the  rectangle  a/38£  will  be  equal 
to  the  space  ADOXX.    Also,  if  this  space  is  found,  the  curve 
is  forthwith  known. 

Many  other  theorems  like  this  could  be  set  down  ;  but 
I  fear  that  these  may  already  appear  more  than  sufficient. 

19.  It    should    be    observed,    however,    that    all    these 
theorems  are  equally  true,  and  can  be  proved    in   exactly 
the  same  way  if  the  curve  AMB  is  convex  to  the  straight 
line  AD. 

20.  Also,  from  what  has  been  shown,  an  easy  method  of 
drawing  curves  (theoretically)   is  obtained,  such   as  admit 
of  measurement  of  some  sort;   in   fact,  you  may  proceed 
thus  :— 

Take  as  you  may  any  right-angled  trapezial  area  (of 
which  you  have  sufficient  knowledge),  bounded  by  two 
parallel  straight  lines  AK,  DL,  a  straight  line  AD,  and  any 
line  KL  whatever;  to  this  let  another  such  area  A  DEC  be 
so  related  that,  when  any  straight  line  FH  is  drawn  parallel 

A 


E  L 


Fig.  162. 


io  DL,  cutting  the  lines  AD,  CE,  KL  in  the  points  F,  G,  H, 
and   some   determinate   straight  line  Z  is  taken,   then   the 


LECTURE  XII  161 

square  on  FH  is  equal  to  the  squares  on  FG  and  Z.  More- 
over, let  the  curve  AIB  be  such  that,  if  the  straight  line  GFI 
is  produced  to  meet  it,  the  rectangle  contained  by  Z  and 
Fl  is  equal  to  the  space  AFGC ;  then  the  rectangle  contained 
by  Z  and  the  curve  AB  is  equal  to  the  space  ADLK.  The 
method  is  just  the  same,  even  if  the  straight  line  AK  is 
supposed  to  be  infinite. 

Example  i.  Let  KL  be  a  straight  line,  then  the  curve 
CGE  is  a  hyperbola.  (Fig.  162.) 

Example  2.  Let  the  line  KL  be  the  arc  of  a  circle  whose 
centre  is  D,  and  let  AK  =  Z;  then  the  curve  AGE  will 
be  a  circle;  and  the  arc  AB  =  AD/2  +  (DL/2AK) .  arc  KL 
(Fig.  163.) 

Example  3.  Let  the  line  KL  be  an  A«-Z-->K 

equilateral   hyperbola,    of  which   the  ^, 

centre  is  A,    and   the  axis   AK  =  Z;        / 

B  D  EL 

then  CGE  will  be  a  straight  line,  and  ,-,.       ,. 

rig.  104* 

the  curve  AB  a  parabola. 

Example  4.  Let  the  line  KL  be  a  parabola,  of  which  the 
axis  is  AD ;  then  the  line  CGE  will  also  be  a  parabola,  and 
the  curve  AB  one  of  the  paraboliforms. 

Example  5.  Let  the  curve  KL  be  an  inverse  or  infinite 
paraboliform  (for  instance,  such  that  FH2  =  Z3/AF);  then 
the  curve  AB  will  be  a  cycloid,  pertaining  to  the  circle 
whose  diameter  is  equal  to  Z.  (See  figure  on  page  164.) 

Perhaps,  if  you  consider,  you  may  think  of  some  examples 

that  are  neater  than  these. 

II 


G\\ 


162  BARROWS  GEOMETRICAL  LECTURES 


NOTE 

The  chief  interest  in  the  foregoing  theorems  lies  in  the 
last  of  all.  The  others  are  mainly  theorems  on  the  change 
of  the  variable  in  integration  (or  rather  that  the  equality 
(oz]ox)  .  Sy  =  (8y/8x) .  Sz  holds  true  in  the  limiting  form  for 
the  purposes  of  integration,  although  of  course  Barrow 
does  not  use  Leibniz'  symbols);  and  secondly,  the  appli- 
cation of  this  principle  to  obtain  general  theorems  on  the 
rectification  of  curves,  by  a  conversion  to  a  quadrature. 
It  must  be  borne  in  mind  that  Barrow's  sole  aim,  expressly 
stated,  was  to  obtain  general  theorems  ;  and  that  he  merely 
introduced  the  cases  of  the  well-known  curves  as  examples 
of  his  theorems ;  and  to  obtain  the  gradient  of  the  tangent 
of  a  curve  in  general  is  the  foundation  of  the  differential 
calculus. 

In  1659,  Wallis  showed  that  .certain  curves  were  capable 
of  "rectification  " ;  the  first-fruits  of  this  was  the  rectification 
of  the  semi-cubical  parabola  by  William  Neil  in   1660,  by 
the   use   of  Wallis'   method.     Almost   simultaneously   this 
curve  was  also  rectified  by  Van  Huraet  (see  Williamson's 
Int.  Cat.,  p.  249)  by  the  use  of  the  geometrical  theorem  :— 
"  Produce  each  ordinate  of  the  curve  to  be  rectified  until 
the  whole  length  is  in  a  constant  ratio  to  the  corresponding 
normal  divided  by  the  old  ordinate,  then  the  locus  of  the 
extremity  of  the  ordinate  so  produced  is  a  curve  whose 
area  is  in  a  constant  ratio  to  the  length  of  the  given  curve." 
Now  this  theorem  is  identical  with  the  theorem  of  §  18; 
hence,  remembering  that  the  semi-cubical  parabola,  whose 
equation  is  R  .y2  =  x3,  is  one  of  those  paraboliforms  of  which 
Barrow  is  so  fond,  and  for  which,  as  we  have  seen,  he  could 
find  both  the  tangent  at  any  point  and  the  area  under  the 
curve  between  any  two  ordinates,  noting  also  the  examples 
given  to  the  theorem  of  §  20,  it  is  beyond  all  doubt  that 
Barrow  must  have  perceived  that  for  this  particular  para- 
boliform  his  curve  OXX  .  .  .  (fig.  160)  was  the  parabola 
4/y2  =  R.(9#  +  4R).     Why  then  did  not  Barrow  give   the 
result?     The  answer,  I  think,  is  given  in  his  own  remark 
before  Problem  IX  in  Appendix  III,  "7  do  not  like  to  put 
my  sickle  into  another  man's  harvest"  where  he  refers  to  the 
work  of  James  Gregory  on  involute  and  evolute  figures. 


LECTURE  XII  163 

Moreover,  this  supposition  may  set  a  date  to  this  section, 
namely  not  before  1659,  an^  not  very  much  later  than 
1 66 1.  For  from  his  opening  remarks  to  the  Appendix  to 
Lect.  XI,  we  can  gather  that  it  was  Barrow's  habit  to  read 
the  work  of  his  contemporaries  as  soon  as  he  could  get 
them,  and  try  to  "go  one  better,"  and  there  are  indications 
enough  in  this  section  to  show  that  Barrow  was  trying  to 
follow  up  the  line  given  by  §  7,  to  obtain  the  reduction  of 
the  problem  of  the  rectification  of  the  parabola  (and  prob- 
ably all  the  paraboliforms  in  general  as  well)  to  a  quadra- 
ture of  some  other  curve;  we  see,  for  instance,  that  he 
obtains  the  connection  between  a  parabolic  arc  and  a 
hyperbolic  area  in  §  7,  and  this  connection  is  obtained 
in  several  other  places  by  different  methods.  He  also 
seeks  general  theorems  in  which  the  quadrature  belongs  to 
one  of  the  paraboliforms  or  the  hyperboliforms  (curves  that 
can  be  obtained  from  a  rectangular  hyperbola  in  the  same 
way  as  the  paraboliforms  are  obtained  from  a  straight  line 
in  Lect.  IX,  §  4) ;  and  the  result  of  using  these  curves, 
whose  general  equation  is  ymxn  —  Rm+n?  is  seen  in  §  20, 
Ex.  5,  where  he  takes  m  =  2,  and  #=i,  and  the  derived 
curve  is  the  Cycloid.  He  does  not  state  that  thus  he  has 
rectified  the  cycloid,  apparently  because  in  Prob.  i,  Ex.  2 
of  App.  Ill,  he  has  obtained  it  in  a  much  simpler  manner 
as  a  special  case  of  another  general  theorem.  (See  critical 
note  that  follows  this  problem.) 

The  great  interest,  however,  of  this  section  centres  in  the 
question  of  the  manner  in  which  Barrow  obtained  the 
construction  for  §  20.  There  is  nothing  leading  to  it  in  any 
theorem  that  has  gone  before  it  in  the  section;  the  only 
case  in  which  he  has  used  the  construction  of  a  subsidiary 
curve,  such  that  the  difference  of  the  squares  on  the 
ordinates  of  the  two  curves  is  constant,  is  in  Lect.  VI, 
§§22,  23,  and  then  his  original  curve  is  a  straight  line.  The 
only  conclusion  that  I  can  come  to  is  that  he  uses  his 
general  theorem  on  rectification  (Lect.  X,  §  5)  analytically 
thus  :— 

If  Z  .  (d§ldx)  —  y,  where  S  is  the  arc  of  the  curve  to  be 
rectified,  and  Y  its  ordinate,  we  must  have  Z  .  (fi\dix)  equal 
to  V(^2  -  Z2),  and  therefore  Z  .  Y  =  \J(y*  -  Z>)dx.  The 
given  construction  is  an  immediate  consequence. 


1 64  BARROW'S  GEOMETRICAL  LECTURES 


Of  course  Barrow  knew  nothing  about  the  notation 
d\[dx  or  [^(yP-T^dx;  his  work  would  have  dealt  with 
small  finite  arcs  and  lines ;  but  the  pervading  idea  is  better 
represented  for  argument's  sake  by  the  use  of  Leibniz' 
notation.  I  suggest  that  Barrow's  proof  would  have  run  in 
something  like  the  following  form  : — 


Draw  JPQR  parallel  and  very 
near  to  IFGH,  cutting  the  curves 
as  shown  in  the  adjoining  diagram, 
and  draw  JT  perpendicular  to  IH  ; 
then 


A     C    K 


L 


Z .  IT  =  area  PFGQ  =  PF  .  FG  ; 

.-.  Z2.  IJ2  =  Z2.  IT2  +  Z2.TJ2  =  PF2.  FG2  +  Z2.  PF2; 

'  .-.  Z.IJ  =  PF.FH. 
Hence,  summing,     Z  .  arc  AB  =  area  ADLK. 

That  Barrow  had,  in  §  20,  Ex.  5,  really  rectified  the 
cycloid  is  easily  seen  from  the  adjoining  diagram.  Barrow 
starts,  I  suppose,  with  the  property  of  the  cycloid  that,  if 
IT,  IM  are  the  tangent  and  normal  at  I,  then  TM  is  per- 
pendicular to  BD.  Let  AD  =  Z, 

then  since  Z.  IT/TN  =  FH,*  we  have  T  A          CK 

FH2  =  Z2.  IT2/TN2  =  Z2 .  TM  .  TN/TN2 
=  Z2 .  TM/TN  =  Z3/AF. 

The  area  under  the  curve  KHL  is 
given  as  proportional  to  the  ordi-    B    M          D  E 
nate  of  what  I  may  call  its  integral 
curve  (see  note  to  Lect.  XI,  §  2),  and  is  easily  shown  to 
be  2AF.FH. 

Hence  arc  Al  =  areaAFHK/Z  -  2AF  .  FH/Z  =  2IT;  that  is, 
equal  to  twice  the  chord  of  the  circle  parallel  to  Tl,  which 
is  also  equal  to  it. 

*  This  follows  at  once  from  the  figure  at  the  top  of  the  page ;  for, 
Z  .  IJ  =  PF .  FH,  and  IT  :  TN  (in  the  lower  figure)  is  equal  to  IJ  :  JT  (in  the 
upper  figure) ;  and  this  is  equal  to  I J  :  PF  or  FH  :  Z  ;  hence,  in  lower  figure 
IT:TN  =  FH  :  Z. 


N     \ 


APPENDIX    I 

Standard  forms  for  integration  of  circular  functions  by 
reduction  to  the  quadrature  of  a  hyperbola. 

Here,  although  it  is  beyond  the  original  intention  to  touch 
on  particular  theorems  in  this  work  ;  *  and  indeed  to  build 
up  these  general  theorems  with  such  corollaries  would  tend 
to  swell  the  volume  beyond  measure;  yet,  to  please  a  friend 
who  thinks  them  worth  the  trouble*  I  add  a  few  observa- 
tions on  tangents  and  secants  of  a  circle,  most  of  which 
follow  from  what  has  already  been  set  forth. 


Fig.  1 66.  .        Fig.  167. 

GENERAL  FOREWORD.  —  Let  ACB  be  a  quadrant  of  a 
circle,  and  let  AH,  BG  be  tangents  to  it;  in  HA,  AC  pro- 
duced take  AK,  CE  each  equal  to  the  radius  CA ;  let  the 

*  Observe  the  words  of  the  opening  paragraph  which  I  have  italicized. 


1 66  BARROWS  GEOMETRICAL  LECTURES 

hyperbola  KZZ  be  described  through  K,  with  asymptotes 
AC,  CZ ;  and  let  the  hyberbola  LEO  be  described  through 
E,  with  asymptotes  BC,  BG. 


Also  let  an  arbitrary  point  M  be  taken  in  the  arc  AB, 
and  through  it  draw  CMS  cutting  the  tangent  AH  in  S,  MT 
touching  the  circle,  MFZ  parallel  to  BC,  and  MPL  parallel 
to  AC.  Lastly,  let  a/3  =  arc  AB,  a/x  =  arc  AM  ;  let  the 
straight  lines  ay,  £/MT^  be  perpendicular  to  a/?;  and  let 
ay  =  AC,  /*£  =  AB,  fu/r  =  08,  and  ^  =  MP. 

1.  The  straight  line  CS  is  equal  to  FZ;  thus  the  sum  of 
the  secants  belonging  to  the  arc  AM,  applied  to  the  line  AC, 

s  equal  to  the  hyperbolic  space  AFZK. 

2.  The  space  a/x£,  that  is,  the  sum  of  the  tangents  to  the 
arc  AM,  applied  to  the  line  a//,,  is  equal  to  the  hyperbolic 
space  AFZK. 

3.  Let  the  curve  AXX  be  such  that  PX  is  equal  to  the 
secant  CS  or  CT;  then  the  space  ACPX,  that  is  the  sum  of 
the  secants  belonging  to  the  arc  AM,  applied  to  the  line  CB, 
is  twice  the  sector  ACM. 


LECTURE  XII— APPENDIX  I        167 

4.  Let   CVV  be  a  curve  such  that   PV  is  equal  to  the 
tangent  AS;  then  the  space  CVP,  that  is,  the  sum  of  the 
tangents  belonging  to  the  arc  AM,  applied  to  the  straight 
line  CB,  is  equal  to  half  the  square  on  the  chord  AM.* 

5.  Let   CQ   be   taken  equal   to  CP,  and  QO  be  drawn 
parallel  to  CE,  meeting  the  hyperbola  LEO  in  0 ;  then  the 
hyperbolic  space  PLOQ  multiplied  by  the  radius  CB  (or  the 
cylinder  on  the  base  PLOQ  of  height  CB)  is  double  the  sum 
of  the  squares  on  the  straight  lines  CS  or  PX,  belonging  to 
the  arc  AM,  and  applied  to  the  straight  line  CB.* 

6.  Hence  the  space  ayi^/x,,  that  is,  the  sum  of  the  secants 
of  the  arc  AM  applied  to  the  line  a/3,  is  equal  to  half  the 
hyperbolic  space  PLOQ.* 

7.  All  the  squares  on  the  straight  lines  fjnf/,  applied  to 
a/x,  are  equal  to  CA  .  CP  .  PX,  that  is,  equal  to  the  parallele- 
piped on  the  rectangular  base  APCD  whose  altitude  is  CS. 

8.  Let  the  curve  AYY  be  such  that   FY  =  AS ;   then,  if 
a  straight  line  Yl  is  drawn  parallel  to  AC,  the  space  ACIYYA 
(that  is,  the  sum  of  the  tangents  belonging  to  the  arc  AM, 
applied  to  the  straight  line  AC,  together  with  the  rectangle 
FCIY)  is  equal  to  half  the  hyperbolic  space  PLOQ.* 

9.  Let   ERK   be  an  equilateral    hyperbola    (that   is,  one 
having  equal  axes),  and  let  the  axes  be  CED,  Cl ;  also  let 
Kl,  KD  be  ordinates  to  these;  let  EVY  be  a  curve  such  that, 

*  These  theorems  are  not  at  first  sight  of  any  great  interest ;  they 
appear  only  to  be  a  record  of  Barrow's  attempts  to  connect  the  quadrature 
of  the  hyperbola  in  some  way  with  the  circle.  But  later,  when  we  find 
that  Barrow  has  the  area  under  the  hyperbola,  their  importance  becomes 
obvious.  (See  critical  note  following  App.  Ill,  Probs.  3,  4.) 


i68  BARROW'S  GEOMETRICAL  LECTURES 

when  any  point  R  is  taken  at  random  on  the  hyperbola, 
and  a  straight  line  RVS  is  drawn  parallel  to  DC,  then  8R, 
CE,  SV  are  in  continued  proportion ;  join  CK ;  then  the 
space  CEYI  will  be  double  the  hyperbolic  sector  DOE. 

10.  Returning   now   to   the   circular  quadrant  ACB,   let 
CE  =  CA ;  and  with  axis  AE,  and  parameter  also  equal  to 
AE,  let  the  hyperbola  EKK  be  described  ;  now  let  the  curve 
AYY  be  supposed  to  be  such  that,  when  any  ordinate  MFY 
is  drawn,  FY  is  equal  to  the  tangent  AS;  draw  YIK,  cutting 
CZ  in  I  and  the  hyperbola  in  K,  and  join  CK,  then  the  space 
ACIYA  is  double  the  hyperbolic  sector  ECK. 

1 1.  COROLLARY. — Hence,  if  with  pole  E,  a  chord  CB,  and 
a  sagitta  CA,  a  conchoid  AW   is  described;   and  if  YFM 
produced  meets  it   in    V;   then   MV •  =  FY ;   and  thus  the 
space  AMV  is  equal  to  the  space  AFY. 

12.  Whence  the  dimensions  of  conchoidal  spaces  of  this 
kind  become  known. 

13.  Let  AE  be  a  straight  line  perpendicular  to  RS  (cutting 
it  in  C) ;  and  let  CE  =  CA  ;  let  AZZ,  EYY  be  two  conchoids, 
conjugate  to  one  another,  described  with  the  same  pole  E 
and  a  common  chord  RS;  from  E  draw  any  straight  line 
EYZ,  cutting  EYY,  AZZ,  RS  in  the  points  Y,  Z,   I ;  also  let 
EKK  be  an  equilateral  hyperbola,  with  centre  C  and  semi- 
axis  CE ;  draw  IK  parallel  to  AE  and  join  CK. 

Then  the  four-sided  space,  bounded  by  AE,  YZ,  and  the 
conchoidal  arcs  EY,  AZ  is  equal  to  four  times  the  hyper- 
bolic sector  ECK. 


LECTURE  XII— APPENDIX  I        169 

14.  We  will  also  add  to  these  the  following  well-known 
measurement  of  cissoidal  space. 

Let  AM B  be  a  semicircle  whose  centre  is  C,  and  let  the 
straight  line  AH  touch  it ;  and  let  AZZ  be  the  cissoid  that 
is  congruent  to  it,  having  this  property,  that,  if  any  point  M 
is  taken  in  the  circumference  AMB,  and  through  it  the 
straight  line  BMS  is  drawn  (cutting  AH  in  8),  and  also  a 
straight  line  MFZ,  cutting  the  cissoid  AZZ  in  Z,  MZ  =  AS ; 
then  in  a  straight  line  aft  take  a/x  equal  to  the  arc  AM,  and  to 
a/z  let  straight  lines  /x,£  be  applied  perpendicular,  and  equal 
to  the  versines  AF  of  the  arc  AM.  Then  the  trilinear  space 
MAZ  is  double  the  space  a/x£.  Hence,  since  the  dimensions 
of  the  space  a//,£  are  generally  known,  and  indeed  can  be 
easily  deduced  from  the  preceding  theorems,  therefore  the 
dimension  of  the  cissoidal  space  MAZ  is  obtained.  Anyone 
may  make  the  calculation  who  wishes  to  do  so. 

The  following  rider  will  close  this  appendix. 

15.  Let  ACB  be  a  quadrant  of  a  circle,  and  let  AH,   BG 
touch  the  circle;  also  let  the  curves  KZZ,   LEO  be  hyper- 
bolas, the  same  as  those  that  have  been  used  above;   let 
the  arc  AM  be  taken,  and  let  it  be  supposed  to  be  divided 
into  parts  at  an  infinite  number  of  points  N  ;  through  these 
draw  radii  ON,  and  let  the  straight  lines  NX  (drawn  parallel 
to  AH)  meet  them  in  the  points  X.     Then  the  sum  of  the 
straight  lines  NX  (taken  along  the  radii)  will  equal  to  the 
space  AFZK/(radius),  and  the  sum  of  the  straight  lines  NX 
(taken  along  parallels  to   AH)  will  be  equal  to  the  space 
PLQO/(3.  radius). 


APPENDIX    II 

Method  of  Exhaustions.     Measurement  of  conical  surfaces. 

For  the  sake  of  brevity  combined  with  clearness,  and 
especially  for  the  latter,  the  proofs  of  the  preceding 
theorems  have  been  given  by  the  direct  method;  by 
which  not  only  is  the  truth  firmly  established,  but  also 
its  origin  appears  more  clearly.  But  for  fear  anyone,  less 
accustomed  to  arguments  of  this  nature,  should  hesitate 
to  use  them,  we  will  add  a  few  examples  by  which  such 
arguments  may  be  made  sure,  and  by  the  help  of  which 
indirect  proofs  of  the  propositions  may  be  worked  out. 

1.  Let  the  ratios  A  to  X,  B  to  Y,  C  to  Z,  be  any  ratios, 
each  greater  than  some  given  ratio  R  to  8 ;  then  will  the 
ratio  of  all  the  antecedents  taken  together  to  all  the  con- 
sequents taken  together  be  greater  than  the  ratio  R  to  8. 

2.  Hence  it  is  evident  that,  if  any  number  of  ratios  are 
each  of  them  greater  than  any  ratio  that  can  be  assigned, 
then  the  sum  of  the  antecedents  bears  a  greater  ratio  to 
the  sum  of  all  the  consequents  than  any  ratio  that  can  be 
assigned. 


LECTURE  XII— APPENDIX  II        171 

3.  Let  ADB  be  any  curve,  of  which  the  axis  is  AD,  and 
to  this  the  straight  line  BD  is  applied;  also  let  the  straight 
line  BT  touch  the  curve,  and  let  BP  be  an  indefinitely  small 
part  of  the  line  BD ;  draw  PO  parallel  to  DT,  cutting  the 
curve  in  N.     Then  I  say  that  PN  will  bear  to  NO  a  ratio 
greater  than  any  assignable  ratio,  R  to  8,  say. 

4.  Hence,  if  the   base    BD   is  divided  into  an  infinite 
number  of  equal  parts  at  the  points  Z,  and  through  these 
points  are  drawn  straight  lines  parallel  to  DA,  cutting  the 
curve  in  E,  F,  G  ;  and  through  the  latter  are  drawn  the 
tangents  BQ,  ER,  F8,  GT,  meeting  the  parallels  ZE,  ZF,  ZG, 
DA  in  the  points  Q,  R,  8,  T;  then  the  straight  line  AD  will 
bear  to  all  the  intercepts  EQ,  FR,  GS,  AT  taken  together  a 
ratio  greater  than  any  assignable  ratio. 

5.  Among  the  results  of  this  we  have  : — 

All  the  lines  EQ,  FR,  GS,  AT  taken  together  are  equal 
to  zero. 

The  lines  ZE,  ZQ ;  ZF,  ZR ;  etc.,  are  equal  to  one 
another  respectively. 

Also  the  small  parts  of  the  tangents  BQ,  ER,  etc.,  are 
equal  to  the  corresponding  small  parts  of  the  curve,  BE,  EF, 
etc.;  and  they  can  be  considered  as  coincident  with  one 
another. 

Moreover,  one  may  safely  assume  anything  which 
evidently  is  consistent  with  these. 

6.  Again,  let  AB  be  any  curve,  of  which  the  axis  is  AD, 
and  let  DB  be  applied  to  it ;  also  let  DB  be  divided  into 
an    indefinite    number    of    equal    parts   at   the   points   Z; 


172  BARROW'S  GEOMETRICAL  LECTURES 

through  these  points  draw  straight  lines  parallel  to  AD, 
cutting  the  curve  in  the  points  X,  and  let  these  be  met 
by  straight  lines  ME,  NF,  OG,  PH,  drawn  through  the  points 

K      AX       R 


H 

Z 

X 
G 

Z 

X 

f 

0 

N 
P 

X 

F 

Z 

\ 

X 

E 

2 

\ 

X 

Z 

\ 

L         D                                   B 

Fig.  176. 

X  parallel  to  BD;  also  let  the  figure  ADBMXNXOXPXRA, 
circumscribed  to  the  segment  ADB  (contained  by  the 
straight  lines  AD,  DB  and  the  curve  AB),  be  greater  than 
any  space  8;  then  I  say  that  the  segment  ADB  is  not  less 
than  the  space  S. 

7.  Also   if    it    is    supposed    that    the    inscribed    figure 
HXGXFXEXZDH    is   less   than   any   space   S;    then   I   say 
that  the  segment  ADB  is  not  greater  than  S. 

8.  Hence,  if  there  is  any  space,  S  say,  the  figure  circum- 
scribed to  which  is  equal  to  the  figure  ADBMNOPRA,  and 
also    the   figure    inscribed    to    it   is    equal   to   the    figure 
HGFEZDH  ;  then  the  space  8  will  be  equal  to  the  segment 
ADB.     For,  as  has  just  been  shown,  it  cannot  be  greater 
than  it,  nor  can  it  be  less. 

Also  these  things  can  be  altered  to  suit  other  modes  of 
circumscription  and  inscription ;  it  should  be  sufficient  to 
have  just  made  mention  of  this. 


LECTURE  XII— APPENDIX  II        173 

NOTE. — In  §  6,  Barrow  uses  the  usual  present-day  method 
of  translating  the  error  for  each  rectangle  across  the  diagram 
to  sum  them  up  on  the  last  rectangle ;  another  point  of 
interest  is  the  striking  similarity  between  the  figure  used 
by  Barrow  and  the  figure  used  by  Newton  in  Lemma  II  of 
Book  I  of  the  Principia,  especially  as  Newton  uses  the 
four-part  division  of  his  base,  which  is  usual  with  Barrow, 
whereas  in  this  place  Barrow,  strangely  for  htm,  uses  a 
five-part  division  of  the  base. 


METHOD  OF  MEASURING  THE  SURFACE  OF  CONES 

Let  AMB  be  any  curve,  whose  axis  is  AD,  and  C  a  given 
point  in  it,  BD  a  straight  line  at  right  angles  to  it.  Any 
point  M  in  the  curve  being  taken,  draw  ME  touching  the 
curve,  and  from  C  draw  CG  perpendicular  to  ME;  also  let 
CV  be  a  straight  line  of  given  length,  perpendicular  to  the 
plane  ADB ;  join  VG.  Then  VG  will  be  perpendicular  to 
MG.  Also  let  RS  be  a  line  such  that,  if  a  straight  line  MIX 
is  drawn  parallel  to  AD,  cutting  the  ordinate  BD  in  I,  and 
the  line  RS  in  X,  then  MP  :  ME  =  VG  :  IX  ;  or,  if  the  line 
AL  is  such  that,  when  MPY  is  drawn  parallel  to  BD,  cutting 
the  axis  AD  in  P,  and  the  line  AL  in  Y,  then  PE  :  ME 
=  VG  :  PY;  then  will  either  of  the  spaces  BRSD  or  ADL 
be  double  the  surface  of  the  cone  formed  by  straight  lines 
through  V  that  move  along  the  curve  AMB. 

Example. — Let  the  curve  AMB  be  an  equilateral  hyper- 
bola, of  which  the  centre  is  C,  and  let  CV  =  CA  =  r,  and 
CP  =  x  (for  it  helps  matters  in  most  cases  to  use  a  calcula- 
tion of  this  kind);  join  MC;  then  the  rectangle  BRSD  is 
double  the  area  AMBV  of  the  cone. 

This  elegant  example  was  furnished  by  that  most  excellent 


1/4  BARROW'S  GEOMETRICAL  LECTURES 

man,  of  outstanding  ability  and  knowledge,  Sir  Francis 
Jessop,  Kt.,  an  Honorable  ornament  of  our  college,  of 
which  he  was  once  a  Fellow-commoner ;  I  shall  venture  to 
adorn  my  book,  as  with  a  jewel,  not  indeed  at  his  request, 
nor  yet  I  hope  against  his  wish,  by  means  of  his  cleverly 
written  work  on  this  matter,  kindly  communicated  to  me. 

PROPOSITION  i 

If  from  a  point  E  in  the  axis  km  of  a  right  cone  ABC/,  a 
straight  line  of  unlimited  length,  EC,  passes  through  the  sur- 
face of  the  cone,  and  if  with  the  end  E  kept  at  rest,  the  line 
EC  is  carried  round  until  it  returns  to  the  place  from  which 
it  started,  so  that  always  some  part  of  it  cuts  the  surface 
of  the  cone  (say,  through  the  hyperbola  CFD  and  the 
straight  lines  DA,  AC  situated  in  the  surface  of  the  cone), 
the  solid  contained  by  the  surface  or  surfaces  generated  by 
the  straight  line  EC  so  moved  and  by  the  portion  of  the 
surface  of  the  cone  bounded  by  the  line  or  lines  CFD,  DA, 
AC,  which  the  straight  line  EC  describes  in  the  surface  as 
it  is  carried  round,  will  be  equal  to  the  pyramid  of  which 
the  altitude  is  equal  to  E«,  the  perpendicular  drawn  from 
the  point  E  to  the  side  of  the  cone,  and  base  equal  to  that 
part  of  the  conical  surface  bounded  by  the  line  or  lines 
CFD,  AD,  AC,  generated  by  the  motion  of  the  line  EC. 

PROPOSITION  2 

Let  ABC/  be  a  right  cone ;  let  it  be  cut  by  the  plane 
CFD  parallel  to  its  axis  km;  let  the  straight  lines  AC,  AD 
be  drawn  from  the  vertex  of  the  cone  to  the  hyperbolic 
line  CFD;  and  upon  the  triangle  ACD  let  the  pyramid  EACD 


LECTURE  XII— APPENDIX  II        175 

be  erected,  having  its  vertex  E  in  the  axis  of  the  cone ;  and 
let  ES  be  perpendicular  to  the  plane  ACD  and  E«  to  the 
side  of  the  cone.  Then  I  say  that  the  conical  surface 
bounded  by  the  hyperbolic  line  CFD  and  the  straight  lines 
DA,  AC  is  to  the  pyramid  EACD  on  the  base  ACD  as  the 
altitude  of  the  pyramid  ES  is  to  the  perpendicular  E^. 

PROPOSITION  3 

Let  ABC/  be  a  given  right  cone;  let  it  be  cut  by  a  plane 
(say,  in  the  triangle  qrt)  and  let  this  plane  cut  the  axis  of 
the  cone  produced  beyond  the  vertex  in  the  point  q\  also 
let  the  common  intersection  of  it  and  the  surface  of  the 
cone  be  the  hyperbolic  line  rSt,  and  let  straight  lines 
kr,  kt  be  drawn  from  A  the  vertex  of  the  cone,  from  the 
point  q  a  perpendicular  q\  to  the  side  A/  of  the  cone 
produced,  and  from  the  point  A  a  perpendicular  AZ  to  the 
plane  qrt.  Then  I  say  that  the  conical  surface,  bounded 
by  the  hyperbolic  line  rst  and  the  straight  lines  rA,  /A,  is 
to  the  hollow  hyperbolic  figure  qr§tq  as  the  perpendicular 
AZ  is  to  the  perpendicular  ^X. 

PROPOSITION  4 

Let  AB/$£"  be  a  given  right  cone;  and  let  it  be  cut  by  a 
plane  HFEG  passing  through  the  axis  below  the  vertex; 
from  the  point  H,  where  the  plane  cuts  the  axis  of  the  cone, 
let  HK  be  drawn  perpendicular  to  any  side  of  the  cone,  and 
from  the  vertex  A  a  perpendicular  AL  to  the  plane  HFEG. 
Then  I  say  that  the  conical  surface,  bounded  by  the  lines 
FEG,  GA,  AF  is  to  the  plane  HGEF  as  the  perpendicular  AL 
is  to  the  perpendicular  HK. 


APPENDIX    III 

Quadrature  of  the  hyperbola.  Differentiation  and  Inte- 
gration of  a  logarithm  and  exponential.  Further  standard 
forms. 

On  looking  over  the  preceding,  there  seems  to  me  to  be 
some  things  left  out  which  it  might  be  useful  to  add. 
Anyone  can  easily  deduce  the  proofs  from  has  already  been 
given,  and  will  obtain  more  profit  from  them  thereby. 

PROBLEM  i 

Let  KEG  be  any  curve  of  which  the  axis  is  AD,  and  let 
A  be  a  given  point  in  AD ;  find  a  curve,  LMB  say,  such  that, 
when  any  straight  line  PEM  perpendicular  to  the  axis  AD 
cuts  the  curve  KEG. in  E  and  the  curve  LMB  in  M,  and  AE 
is  joined,  and  TM  is  a  tangent  to  the  curve  LMB,  then  TM 
shall  be  parallel  to  AE. 

The  construction  is  made  as  follows :  —Through  any 
point  R,  taken  in  the  axis  AD,  draw  a  straight  line  RZ 
perpendicular  to  AD ;  let  EA  produced  meet  it  in  8,  and 
in  the  straight  line  EP  take  PY  equal  to  RS;  in  this  way 
the  nature  of  the  curve  OYY  is  determined;  then  let  the 
rectangle  contained  by  AR  and  PM  be  equal  to  the  space 
AYYP  (or  PM  is  equal  to  the  space  AYYP/AR).  Then  the 
curve  AYYP  shall  have  the  proposed  property. 

It  should  also  be  easily  seen  that,  other  things  remaining 


LECTURE  XII— APPENDIX  III       17 7 

the  same,  if  the  curve  QXX  is  such  that,  if  EP  cuts  it  in  X, 
PX  =  AS ;  then  the  space  AXXP  is  equal  to  the  rectangle  con- 
tained by  AR  and  the  arc  LM,  or  space  AXXP/AR  =  arc  LM. 

Example  i. — Let  ADG  be  a  quadrant  of  a  circle;  if  EP 
is  any  straight  line  perpendicular  to  AD,  join  DE.  It  is 
required  to  draw  the  curve  AMB  such  that,  if  EPM  produced 
meets  it  in  M,  and  MT  touches  the  curve,  then  MT  shall  be 
parallel  to  DE. 

The  construction  is  as  follows: — Draw  AZ  parallel  to 
DG,  and  let  DE  produced  meet  it  in  S ;  let  the  curve  AYY 
be  such  that,  if  PE  produced  meets  it  in  Y,  PY  =  AS;  then 
take  PM  =  space  AYP/AD ;  and  the  construction  is  effected. 

NOTE.— If  the  curve  QXX  is  such  that  PX  =  D8  (or  if 
AQ  =  AD  and  QXX  is  a  hyperbola  bounded  by  the  angle 
ADG),  then  arc  AM  .  AD  =  space  AQXP. 

Example  2. — Let  AEG  be  any  curve  whose  axis  is  AD 
such  that,  when  through  any  point  E  taken  in  it  a  straight 
line  EP  is  drawn  perpendicular  to  AD,  and  AE  is  joined,  then 
AE  is  a  given  mean  proportional  between  AR  and  AP  of  the 
order  whose  exponent  is  n\m.  It  is  required  to  find  the 
curve  AMB,  of  which  the  tangent  TM  is  parallel  to  AE. 

Observe  about  the  curve  AM  that  n  :  m  =  AE  :  arc  AM. 

Now,  if  nfm  =  1/2  (or  AE  is  the  simple  geometrical  mean 
between  AR  and  AP),  then  AEG  will  be  a  circle,  and  AMB 
the  ordinary  cycloid.  Hence  the  measurement  of  the  latter 
comes  out  from  a  general  rule. 

These  also  follow  from  the  more  general  theorem  added 
below. 

12 


178  BARROW'S  GEOMETRICAL  LECTURES 


NOTE 

At  first  sight  the  foregoing  proposition,  stated  in  the 
form  of  a  problem,  but  (by  implication  in  the  note  above) 
referred  to  by  Barrow  as  a  theorem,  would  appear  to  be  an 
attempt  at  an  inverse-tangent  problem.  But  "  the  sting  is 
in  the  tail " ;  this,  and  most  of  those  which  follow,  are  really 
further  attempts  to  rectify  the  parabola  and  other  curves, 
by  obtaining  a  quadrature  for  the  hyperbola.  That  this  is 
so  is  fairly  evident  from  the  note  to  Ex.  i  above ;  and  it 
becomes  a  moral  certainty  when  we  come  to  Problem  IV, 
where  Barrow  is  at  last  successful. 

The  first  sentence  of  the  opening  remark  to  this 
Appendix,  which  I  have  put  in  italics,  makes  it  certain  that 
these  were  Barrow's  own  work.  The  reference  to  Wallis  at 
the  end  of  Problem  IV  almost  "shouts"  the  fact  that  it  was 
through  reading  Wallis'  work  that  Barrow  began  to  accumu- 
late, as  was  his  invariable  practice,  a  collection  of  general 
theorems  connecting  an  arc  with  an  area ;  it  is  also  probable 
that  it  was  only  just  before  publication  that  he  was  able  to 
complete  his  collection  with  the  proof  that  the  area  under 
a  hyperbola  was  a  logarithm. 

The  proof,  as  Barrow  states,  for  the  construction  given 
in  Problem   I  is  very  easily  made  out,  by  drawing  another 
ordinate  NFQY  parallel  and  near  to  MEPY  and  MW  parallel 
to  PQ  to  cut  NQ  in  W.     For  we  have  then 
PE/PA  -  RS/AR  -  PY/AR  =  MW/NW  =  MP/PT,  .'.  AE//MT. 

Example  i  is  not  truly  an  example  of  the  problem  ;  if  we 
allow  for  Barrow's  inversion  of  the  figure  (a  bad  habit  of 
his  that  probably  caused  trouble  to  his  readers),  to  render 
this  a  true  example  of  the  method  of  the  problem,  AD  .  PM 
should  be  made  equal  to  the  space  DYP  instead  of  the 
space  AYP ;  this,  however,  would  have  made  the  curve  lie 
on  the  same  side  of  AD  as  the  quadrant,  at  an  infinite 
distance-,  so  Barrow  subtracts  the  infinite  constant,  equal 
to  the  area  QADY,  and  thus  gets  a  curve  lying  on  the  other 
side  of  the  line  AD,  fulfilling  the  required  conditions. 
Example  2,  however,  is  a  true  example  of  the  problem  ; 
and  it  is  particularly  noteworthy  on  account  of  the  fact 
that  it  rectifies  the  cycloid,  a  result  previously  attained  in 


LECTURE  XII—  APPENDIX  III       179 

Lect.  XII,  §  20,  Ex.  5  ;  but,  as  has  been  noted,  the  matter 
is  not  so  clearly  put  in  that  as  it  is  here  ;  for,  in  this 
Example  2,  since  AE2  =  AP.  AR,  the  curve  AEG  is  evidently 
a  circle,  and  it  follows  from  the  property  that  the  tangent 
at  M  is  parallel  to  AE,  that  the  curve  AMB  is  the  cycloid; 
the  theorem  states  that  the  arc  AM  is  equal  to  twice  the 
chord  AE;  and  thus  Barrow  has  undoubtedly  rectified  the 
cycloid,  and  thus  anticipated  Sir  C.  Wren,  who  published 
his  work  in  the  Phil.  Trans,  for  1673.  Moreover,  and 
Barrow  seems  to  be  prouder  of  this  fact  than  anything  else, 
Barrow's  theorem  is  a  general  theorem  for  the  rectification 
of  all  curves  of  the  form  given  by 

X  =  2a  cosmln  0,     Y  =  2am/n  .  \\siri*  0  cos(m~n]ln  0  dO. 

If  m/n  =  2,  the  curve,  as  Barrow  remarks,  is  a  cycloid  ; 
this  is  also  evident  analytically  if  the  equations  above  are 
worked  out.  If,  however,  m/n  is  equal  to  any  odd  integer, 
the  curve  AEG  has  a  polar  equation  r  =  a  cos28  0,  and  the 
curve  AMB  is  one  of  the  form  given  by  the  equations 

X  =  «rttf*  +  10,     Y  =  a 


and  this,  in  the  particular  case  when  s  =  i,  is  the  three- 
cusped  hypocycloid,  X2/3  +  Y2/3  =  a2/s,  and  the  arc  of  this 
curve  is  given  as  3AE/2  (for  my  n  is  Barrow's  m  —  n),  or 
3  al/3x213  ;  and  thus  the  theorem  also  rectifies  the  three- 
cusped  hypocycloid;  though,  of  couise,  Barrow  does  not 
mention  this  curve,  nor  can  I  see  a  simple  theorem  by  which 
Barrow  could  have  performed  the  integration,  denoted  by 
jsin2  0  cos  0  d6,  by  a  geometrical  construction. 

PROBLEM  2 

To  draw  a  curve,  AMB  say,  of  which  the  axis  is  AD,  such 
that,  any  point  M  being  taken  in  it,  if  MP  is  drawn  perpen- 
dicular to  AD  and  MT  is  supposed  to  be  a  tangent  to  the 
curve,  then  TP  :  PM  shall  be  an  assigned  ratio. 

Let  any  straight  line  R  be  taken  ;  find  PY,  such  that 
TP  :  PM  (which  ratio  the  assigned  relation  will  give)  is  equal 
to  the  ratio  R  :  PY  (and  this  is  to  be  taken  along  the  line 


i8o  BARROWS  GEOMETRICAL  LECTURES 

PM  and  at  right  angles  to  the  axis  AD) ;  and  through  the 
points  Y  obtained  in  this  way  let  the  curve  YYK  be  drawn  ; 
then,  if  PM  is  made  equal  to  the  space  APY/R,  the  nature 
of  the  curve  AMB  will  be  established. 

Example  i. — Let  ADG  be  a  quadrant  of  a  circle,  of  which 
the  radius  is  equal  to  the  assigned  length  R ;  let  it  be 
desired  that  the  ratio  of  TP  to  PM  shall  be  equal  to  that  of 
R  to  arc  AE ;  then,  since  as  prescribed,  R  :  arc  AE  =  R  :  PY  ; 
PY  =  arc  AE;  and  hence  PM  =  APY/R. 

Example  2.  — Let  ADG  be  a  quadrant  of  a  circle,  and 
suppose  that  the  ratio  TP :  PM  has  to  be  equal  to  that  of 
PE  :  R ;  then  PY  will  be  equal  to  the  tangent  of  the  arc 
GE;  and  the  space  APYY  is  equal  to  R  .  arc  AE.  Then 
PM  =  arc  AE. 

PROBLEM  3 

Being  given  any  figure  AMBD  whose  axis  is  AD  and  base 
DB,  it  is  required  to  find  a  curve  KZL  such  that,  when  any 
straight  line  ZPM  is  drawn  parallel  to  DB,  cutting  AD  in  P, 
and  it  is  supposed  that  ZT  touches  the  curve  KZL,  then 
TP  =  PM. 

The  construction  is  as  follows : — 

Let  OYY  be  a  curve  such  that,  any  finite  straight  line  R 
being  taken,  and  PMY  produced,  PM  :  R  =  R  :  PY ;  then, 
taking  any  point  L  in  BD  produced,  draw  LE  at  right  angles 
to  DL,*  so  that  DL:  R  =  R  :  LE ;  then,  with  asymptotes 
DL,  DG,*  describe  the  hyperbola  EXX  passing  through  E ; 
let  the  space  LEXH  be  equal  to  the  space  DOYP,  and  pro- 


AD  is  produced  to  G  and  LE  is  in  the  same  sense  as  DG. 


LECTURE  XII— APPENDIX  III       181 

duce  XH  and  YP  to  meet  in  Z.  Then  will  Z  be  a  point  in 
the  required  curve,  and  if  ZT  is  a  tangent  to  it,  TP  =  PM. 

It  is  to  be  noted  that,  if  the  given  figure  is  a  rectangle 
ADBC,  the  curve  KZL  has  the  following  property.  DH  is  a 
geometric  mean  between  DL  and  DO  of  the  same  order  as 
DP  is  an  arithmetic  mean  between  DA  and  0  (or  zero). 

Now,  if  any  curve  KZL  is  described  with  this  property, 
and  the  tangent  ZT  is  found  practically,  then  the  hyberbolic 
space  LEXH  will  be  found,  and  this  in  all  cases  is  equal  to 
the  rectangle  contained  by  TP  and  AP.* 

It  can  also  be  seen  that 

(i)  the  space  ADLK  =  R(DL-DO); 

(ii)  the  sum  of  ZP2,  etc.  =  R(DL2  -  D02)/2,  and  the  sum 
of  ZP3,  etc.  =  R(DL3  -  D03)/3,  and  so  on ;  f 

(iii)  if  it  is  supposed  that  <£  is  the  centre  of  gravity  of  the 
figure  ADLK,  and  <f>\p  is  drawn  perpendicular  to  AD  and  </>£ 
to  DL,  then  ^  =  (DL+  DO)/4,  and  <££  =  R -  AD.  DO/LO. 

PROBLEM  4 

Let  BDH  be  a  right  angle,  and  BF  parallel  to  DH ;  with 
DB,  DH  as  asymptotes,  let  a  hyperbola  FXG  be  described 
to  pass  through  F ;  with  centre  D  describe  the  circle  KZL ; 
lastly  let  AM  B  be  a  curve  such  that,  if  any  point  M  is  taken 
in  it,  and  through  M  the  straight  line  DMZ  is  drawn,  and 
it  is  also  assumed  that  Dl  =  DM  and  IX  is  drawn  parallel 


*  Here  Barrow  seeks  the  curve  whose  subtangent  is  constant  and  obtains 
it ;  he,  however,  does  not  at  first  seem  to  perceive  the  exponential  character 
of  it.  For,  although  he  states  the  property  of  the  geometric  and  arithmetic 
means,  it  is  not  till  in  connection  with  the  next  problem  that  he  states  that 
this  has  anything  to  do  with  logarithms. 

f  As  usual,  these  quantities  have  to  be  applied  to  AD. 


1 82  BARROW'S  GEOMETRICAL  LECTURES 

to  BF,  then  the  hyperbolic  space  BFXI  is  equal  to  twice 
the  circular  sector  ZDK.  It  is  required  to  draw  the  tangent 
at  M  to  the  curve  A  MB. 

Draw  DS  perpendicular  to  DM,  and  let  DB  .  BF  =  R2; 
then  make  DK  :  R  -  R  :  P,  and  then  DK  :  P  =  DM  :  DT; 
join  TM  ;  then  TM  will  touch  the  curve  AMB. 

It  is  to  be  observed  that  the  curve  has  the  following 
property.  Dl  is  a  geometric  mean  between  DB  and  DO 
(or  DA)  of  the  same  order  as  the  arc  KZ  is  an  arithmetic 
mean  between  0  (or  zero)  and  the  arc  KL  That  is,  if  Dl 
is  a  number  in  the  geometric  series  beginning  with  DB  and 
ending  with  DA,  and  0,  KL  are  the  logarithms  of  DB,  DA, 
then  KZ  will  be  the  logarithm  of  Dl.  Or,  working  the 
other  way  (the  way  in  which  ordinary  logarithms  go),  if  Dl 
is  a  number  in  the  geometric  series  starting  with  DO  and 
ending  with  DB,  and  0  is  the  logarithm  of  DO,  and  the  arc 
LK  that  of  DB,  then  the  arc  LZ  will  be  the  logarithm 
of  Dl. 

Now,  ii  the  curve  is  completely  drawn  and  the  tangent 
to  it  determined  practically,  it  is  evident  that  the  circular 
equivalent  of  the  hyperbolic  space  is  given,  or  the  hyper- 
bolic equivalent  of  the  circular  sector. 

That  most  eminent  man,  Wallis,  *  worked  out  most  clearly 
the  nature  and  measurement  of  this  Spiral  (as  well  as  of 
the  space  BDA)  in  his  book  on  the  cycloid ;  and  so  I  will 
say  no  more  about  it. 


*  Wallis'  chief  works  connected  with  the  problems  of  Infinitesimal 
Calculus  are  in  course  of  preparation,  and  will  be  issued  shortly ;  so  that 
it  has  not  been  thought  necessary  to  give  here  anything  further  than  this 
reference. 


LECTURE  XII— APPENDIX  III       183 


NOTE 

The  two  foregoing  propositions  are  particularly  interest- 
ing in  their  historical  associations.  Logarithms  had  been 
invented  at  the  beginning  of  the  seventeenth  century,  and  the 
method  of  Briggs  (Arithmetica  Logarithmica^  1624)  was  still 
fresh.  Logarithms  were  devised  as  numbers  which  increased 
in  arithmetical  progression  as  other  numbers  related  to 
them  increased  in  geometrical  progression.  We  know 
that  Wallis  had  evaluated  the  integral  of  a  positive  integral 
power  of  the  variable,  and  later  had  extended  his  work  to 
other  powers  ;  Cavalieri  had  also  obtained  the  same  results 
working  in  another  way ;  also  Fermat  had  used  the  method 
of  arithmetic  and  geometric  means  as  the  basis  of  his 
work  on  integration,  and  he  specially  remarks  that  it  is 
a  logarithmic  method;  but  it  was  left  to  Gregory  St 
Vincent  to  perform  the  one  remaining  integration  of  a 
power  when  the  index  was  -  i.  This  he  did  by  the 
method  of  exhaustions,  working  with  a  rectangular  hyper- 
bola referred  to  its  asymptotes;  he  stated  (in  1647)  that, 
if  areas  from  a  fixed  ordinate  increased  in  arithmetic 
progression,  the  other  bounding  ordinates  decreased  in 
geometric  progression.*  This  is  practically  identical  with 
the  special  type  that  Barrow  takes  as  an  example  to 
Problem  3 ;  but  it  was  left  to  Barrow  to  give  the  result 
in  a  definite  form.  At  the  same  time  we  see  that,  if 
Barrow  owed  anything  at  all  to  Fermat,  we  must  credit 
Fermat's  remark  with  being  the  source  of  Barrow's  ideas 
on  the  application  of  these  arithmetic  and  geometric 
means.  As  usual  with  Barrow,  he  gives  a  pair  of 
theorems,  perfectly  general  in  form,  one  for  polar  and 
the  other  for  rectangular  coordinates.  He  proves  that 
the  area  under  the  hyperbola  referred  to  its  asymptotes, 
included  between  two  ordinates  whose  abscissae  are  #,  b,  is 
log  (b\a\  though  he  is  unaware  apparently  of  the  value  of 
the  base  of  the  logarithm.  I  say  apparently,  because  I 
will  now  show  that  it  is  quite  within  the  bounds  of 
probability  that  Barrow  had  found  it  by  calculation ; 

*  Brouncker  used  the  same  idea  in  1668  to  obtain  an  infinite  series  for 
the  area  under  a  hyperbola. 


1 84  BARROW'S  GEOMETRICAL  LECTURES 


supposing  my  suggestion  is  true,  however,  Barrow  would 
at  that  time  have  been  unable  to  have  proved  his 
calculation  geometrically r,  or  indeed  in  any  other  theoretical 
manner,  and  so  would  not  have  mentioned  the  matter ;  as 
we  see,  he  leaves  the  constant  to  be  determined  practically 
(Mechanice),  this  way  being  just  as  good  in  his  eyes  as  any 
other  that  was  not  geometrical. 

Let  AFB  be  a  paraboliform  such  that  PF 

is  the  first  of  m  -  i  means  between  PG  and        • =£— «B 

P E.    Also  let  V K D  be  another  curve  such  that 

space  AVDP=  R  .  PF. 

Then,  taking  AC  =  CB,  to  avoid  a  con- 
stant, the  equation  to  AFB  is 

PF™  =  AP  .  PG—1, 
and  the  equation  of  VDK  is 


Now  area  LKDP  =  R.(PF-HL) 

/.  LP  .  PD  =  R  .  PG1-1^  .  (AP1/W  -  AL1""). 

Hence,  if  we  put  x  for  AP,  we  obtain  \dx]xl~llm  =  the 
sum  of  m  .  LP  .  PD/R  .  PG1-1'™  =  m  .  (AP1/m  -  AL1^). 

But  if  m  is  indefinitely  increased,  and  R  is  taken  equal 
to  m,  the  curve  VKD  tends  to  become  a  rectangular  hyper- 
bola; and  in  Problems  3,  4,  Barrow  has  shown  that  the 
area  is  proportional  to  log  AP/AL.  Hence  log  AP/AL  is  the 
limiting  value  of  /»(AP1/m  —  AL1/m),  when  m  is  indefinitely 
increased,  or  log  x  is  the  limiting  value  of  (x11  -  i)/«,  when 
n  is  indefinitely  small. 

Now  remembering  that  Briggs  in  his  Arithmetica  Logarith- 
mica  had  given  the  value  of  10  to  the  power  of  i/254  as 
i  '0000000000000001278  191  4932003235,  it  would  not  have 
taken  five  minutes  to  work  out  log  10  =  2*3058509  .  .  .; 
hence,  calling  this  number  /A,  Barrow  has 


\dx\x  =  //,  .  log 


lQ 


Considering  Barrow's  fondness  for  the  paraboliforms,  it 
would  seem  almost  to  be  impossible  that  he  should  not 
have  carried  out  this  investigation;  although,  if  only  for 
his  usual  disinclination  to  "put  his  sickle  into  another 


LECTURE  XII—  APPENDIX  III       185 

man's  harvest,"  as  he  remarks  at  the  head  of  Problem  9, 
he  does  not  publish  it;  he  in  fact  refers  to  Wallis'  work 
on  the  Logarithmic  Spiral  as  a  reason  why  he  should  say 
no  more  about  it.  It  is  to  be  noted  that,  in  Problem  4, 
Barrow  constructs  the  Equiangular  Spiral,  and  then  proves 
it  to  be  identical  with  the  Logarithmic  Spiral.  Hence, 

7/1 

if  r—  =  C,   r  =  ae  and  conversely;    thus  d(ax)jdx  =  \(ax 
dr 

and  ^axdx  =  max,  where  K,  m  have  to  be  determined. 

If  we  do  not  allow  that  Barrow  had  found  out  '  a  value 
for  the  base  of  the  logarithm,  yet  assuming  log  x  to  stand 
for  a  logarithm  to  an  unknown  base,  Barrow  has  rectified 
the  parabola,  effected  the  integration 
of  tan  6,  and  the  areas  of  many  other 
spaces  that  he  has  reduced  to  the  quad- 
rature of  the  hyperbola.  For  instance,  /I 
in  Lect.  XII,  §  20,  Ex.  3,  he  shows  that  / 
Z.arc  AB  =  area  ADLK. 


Now        ATLK  =  Z2  .  log  (^2  .  AT/ZJ/2  +  Z2/4 

.?.  ADLK  =  iZ2./^(AD  +  DL)/Z  +  JAD.D 
In  modern  notation,  since  Z.  DL  =  AD2/2, 

=  Jfl2  .  log  [{x  +  J(x2  +  a*)}  /a] 


r 


where  the  base  of  the  logarithm  has  to  be  determined. 

Similarly,  in  Lect.  XII,  App.  I,  §  2,  he  states  that  the 
sum  of  the  tangents  belonging  to  the  arc  AM  applied  to  the 
line  a//,  is  equal  to  the  hyperbolic  space  AFZK;  that  is, 

f0  tan  0  dO  =  log  AF/AC  =  log  cos  0. 

Jo 

The  theorem  of  §  i  is  the  same  thing  in  another  form. 

Again,  in  Lect.  XII,  App.  I,  §  4,  we  have  the  equivalent 
of  the  integral  of  sin  6 ;  since  Barrow's  integrals  are  all 

CO 

definite,    we   find   it  "in    the   form    I    sin  6  dO  =  2  cos2  0/2. 

From    §  5,    we    obtain    I    sec2  0  d  (sin  6)  =  \      dOlcos  0  or 
k  k 


1 86  BARROWS  GEOMETRICAL  LECTURES 

6 


sec  6  dO  given  as  J  log  {(i  +  sin  0)/(i  -  sin  0)},  which  of 

course  can  be  reduced  immediately  to  the  more  usual 
form  log  tan  (6/2  4-  7r/4)  ;  the  same  result  is  obtained  from 
§§6,  7  ;  or  they  can  be  exhibited  in  the  form  §dx/(a2  -  x2) 
=  [kg  (a  +  x)l(a  -  x)}/2a. 

The  theorem  of  §  8  is  a  variant  of  the  preceding  and 
proves  that  J  cos  6  d(tan  0)  is  equal  to  J  tan  0  d(cos  0)  -  tan 
0  .  cos  0,  both  being  equal  to  J  sec  6  dO. 

The  theorem  of  §  9  reduces  immediately  to  ^dx/fj(x2  +  a2) 


Thus  Barrow  completes  the  usual  standard  forms  for  the 
integration  of  the  circular  functions. 

There  is  one  other  point  worth  remarking  in  this  con- 
nection, as  it  may  account  for  the  rushing  into  print  of  this 
rather  undigested  Appendix  ;  I  have  already  noted  that, 
from  Barrow's  own  words,  this  Appendix  was  added  only 
just  before  the  publication  of  the  book.  I  imagine  this 
was  due  to  Barrow's  inability  to  complete  the  quadrature 
of  the  hyperbola  to  his  rather  fastidious  taste.  But,  in 
1668,  Nikolaus  Kaufmann  (Latine  Mercator)  published  his 
Logarithmotechnia,  in  which  he  gave  a  method  of  finding 
true  hyperbolic  logarithms  (not  Napierian  logarithms);  of 
this  publication  Prof.  Cajori  says:  —  "Starting  with  the 
grand  property  of  the  rectangular  hyperbola  .  .  .,  he 
obtained  a  logarithmic  series,  which  Wallis  had  attempted 
but  failed  to  obtain."  (Rouse  Ball  attributes  the  series  to 
Gregory  St  Vincent.)  This  may  have  settled  any  qualms 
that  Barrow  had  concerning  the  unknown  base  of  his 
logarithms,  and  decided  him  to  include  this  batch  of 
theorems,  depending  solely  on  the  quadrature  of  the 
hyperbola,  and  merely  requiring  a  definite  solution  of  the 
latter  problem  to  enable  Barrow  to  complete  his  standard 
forms.  Kaufmann  obtained  his  series  by  shifting  one  axis 
of  his  hyperbola,  so  that  the  equation  became  y  =  i/(i  +x), 
expanded  by  simple  division,  and  integrated  the  infinite 
series  term  by  term,  thus  obtaining  the  area  measured  from 
the  ordinate  whose  length  was  unity,  and  avoiding  the 
infinite  area  close  to  the  asymptote. 


LECTURE  XII— APPENDIX  III       187 

PROBLEM  5 

Let  EDG  be  any  space  bounded  by  the  straight  lines 
DE,  DG  and  the  curve  ENG,  and  R  any  straight  line  of 
given  length  ;  it  is  required  to  find  a  curve  AMB  such  that, 
when  any  straight  line  DNM  is  drawn  from  D,  and  DT  is 
perpendicular  to  it,  and  MT  touches  the  curve  AMB,  then 
shall  DT:DM  =  R  :  DM.* 

Let  KZL  be  a  curve  such  that  DZ2  =  R  .  DN,  and,  the 
straight  line  DB  being  drawn,  of  arbitrary  length,  let 
DB  :  R  =  R  :  BF,  where  BF,  and  also  DH,  is  at  right  angles 
to  DB.  Then  through  F,  within  the  angle  BDH,  draw  the 
hyperbola  FXX,  and  let  the  space  BFXI  (where  IX  is  supposed 
to  be  parallel  to  BF)  be  equal  to  double  the  space  ZDL; 
lastly,  let  DM  =  Dl.  Then  M  will  be  a  point  on  a  curve 
such  as  is  required ;  and  if  a  straight  line  MT  touches  the 
curve  at  any  point  M,  then  will  TD  :  TM  =  R  :  DN. 

PROBLEM  6 

Again,  let  EDG  be  a  given  space  (as  in  the  preceding) ; 
it  is  required  to  find  a  curve  AMB  such  that,  if  any  straight 
line  DNM  is  drawn,  and  DT  is  perpendicular  to  it,  and  MT 
touches  the  curve,  then  DT  shall  be  equal  to  DN. 

Take  any  straight  line  of  length  R,  and  let  DZ2  =  R3/DN  ; 
also  having  taken  DB  (to  which  DH,  and  BF  (-  R3/DB2), 
are  perpendiculars)  assume  that  through  F  is  drawn,  between 
the  asymptotes  DB,  DH,  a  hyperboliform  of  the  second  kind 
(that  is,  one  in  which  the  ordinates,  as  BF  or  IX,  are  fourth 

*  The  next  four  problems  constitute  Barrow's  conclusion  of  his  work  on 
Integration.  Probs.  5,  6  give  graphical  constructions  for  integration,  and 
7,  8  find  graphically  the  bounding  ordinate  or  radius  vector  for  a  figure 
of  given  area,  i.e.  graphical  differentiation  of  a  kind. 


1 88  BARROW'S  GEOMETRICAL  LECTURES 

proportionals  in  the  ratio  DB  to  R,*  or  Dl  to  R).  Then 
take  the  space  BIXF  equal  to  double  the  space  ZDL;  and 
let  DM  =  Dl ;  then  M  will  be  a  point  on  the  required  curve ; 
and  if  MT  touches  it,  DT  =  DN. 

PROBLEM  7 

Let  ADB  be  any  figure,  of  which  the  axis  is  AD  and  the 
base  is  DB,  and,  any  straight  line  PM  being  drawn  parallel 
to  DB,  let  the  space  ARM  be  given  (or  expressed  in  some 
way) ;  it  is  required  from  this  to  draw  the  ordinate  PM,  or 
to  give  some  expression  for  it. 

Take  any  straight  line  R,  and  let  R  .  PZ  =  space  APM  ; 
in  this  way  let  the  line  AZZK  be  produced;  find  ZO  the 
perpendicular  to  it;  then  PZ :  PO  =  R  :  PM. 

Otherwise.  Take  PZ  =  V(2APM) ;  let  ZO  be  perpendicular 
to  the  curve  AZK ;  then  PM  =  PO. 

PROBLEM  8 

Let  ADB  be  any  figure,  bounded  by  the  straight  lines 
DA,  DB  and  the  curve  AMB,  and  through  D  let  any  straight 
line  DM  be  drawn  ;  given  the  space  ADM,  it  is  required  to 
find  the  straight  line  DM. 

Take  any  straight  line  R,  and  let  DZ  =  2ADM/R ;  draw 
ZO  perpendicular  to  the  curve  AZK ;  let  DH,  the  perpen- 
dicular to  DM,  meet  it;  then  DM2  =  R  .  DO. 

Otherwise.  Let  DZ  =  V(4ADM) ;  and  draw  ZO  per- 
pendicular to  the  curve  AZK ;  let  DH,  the  perpendicular  to 
DZ,  meet  it;  then  DM2  =  DZ  .  DO. 


If  DB  :  R  =  R  :  P  =  P  :  BF,     DB2  :  R2  =  R  :  BF,     or     BF  =  R3/DB2. 


LECTURE  XII— APPENDIX  III       189 

NOTE 

These  four  problems  are  generally  referred  to  by  the  i 
authorities  as  "inverse-tangent"  problems.  I  do  not  think 
this  was  Barrow's  intention.  They  are  simply  the  com-  » 
pletion  of  his  work  on  integration,  giving  as  they  do  a 
method  of  integrating  any  function,  which  he  is  unable  to 
do  by  means  of  his  rules,  by  drawing  and  calculation^ 
Thus,  the  probl  ;m  of  §  5  reduces  to  : — "  Given  any  function, 
f(x)  say,  construct  the  curve  whose  polar  equation  is 
r  =  /(#),  perfoi  m  the  given  construction,  and  the  value 
of  \xQf(x)dx  is  equal  to  R  log  DB/DI  or  R  log  DB/DM." 
Similarly,  in  Problem  6,  the  value  of  ^dxjf(x)  is  given  as 
i /DM  -  i/DB.  The  construction  as  given  demands  the 
next  two  problems,  or  one  of  those  which  follow,  called 
by  Barrow  "evolute  and  involute"  constructions.  As  an 
alternative,  Barrow  gives  an  envelope  method  by  means  of 
the  sides  of  the  polar  tangent  triangle.  It  is  rather  re- 
markable that  as  Barrow  had  gone  so  far,  he  did  not  give 
the  mechanical  construction  of  derivative  and  integral 
curves  in  the  form  usual  in  up-to-date  text-books  on 
practical  mathematics,  which  depend  solely  on  the  property 
that  differentiation  is  the  inverse  of  integration. 

With  regard  to  the  propositions  that  follow  under  the 
name  of  problems  on  "evolutes  and  involutes,"  it  must  be 
noted  that,  although  at  first  sight  Barrow  has  made  a 
mistake,  since  the  involute  of  a  circle  is  a  spiral  and  cannot 
under  any  circumstances  be  a  semicircle;  yet  this  is  not 
a  mistake,  for  Barrow's  definition  of  an  involute  (whether 
he  got  it  from  James  Gregory's  work  or  whether  he  has 
misunderstood  Gregory)  is  not  the  usual 
one,  but  stands  for  a  polar  figure  equivalent  A 
in  area  to  a  given  figure  in  rectangular  coor- 
dinates, and  vice  versa.  In  a  sense  somewhat 
similar  to  this  Wallis  proves  that  the  circular 
spiral  is  the  involute  of  a  parabola. 

Hence,    these    problems   give    alternative 
methods  for  use  in  the  given  constructions  for  Problems  5,  6. 
Thus  in  the  adjoining  diagram,  it  is  very  easily  shown  that 
area  D/*/xB  =  £  area  DBMP. 


190  BARROWS  GEOMETRICAL  LECTURES 

That  brilliant  geometer,  Gregory  of  Aberdeen,  has  set 
on  foot  a  beautiful  investigation  concerning  involute  and 
evolute  figures.  I  do  not  like  to  put  my  sickle  into  another 
man's  harvest,  but  it  is  permissible  to  interweave  amongst 
these  propositions  one  or  two  little  observations  pertaining 
in  a  way  to  such  curves,  which  have  obtruded  themselves 
upon  my  notice  whilst  I  have  been  working  at  something  else. 


PROBLEM  9 

Let  ADB  be  any  given  figure,  of  which  the  axis  is  AD 
and  the  base  is  DB  ;  it  is  required  to  draw  the  evolute 
corresponding  to  it. 

With  centre  C,  and  any  radius  CL,  let  a  circle  LXX  be 
described;  also  let  KZZ  be  a  curve  such  that,  when  any 
line  MPZ  you  please  is  drawn  parallel  to  BD,  the  rectangle 
contained  by  PM  and  PZ  is  equal  to  the  square  on  CL 
(or  PZ  is  equal  to  CL2/PM).  Then  let  the  arc  LX  =  space 
DKZP/CL  (or  sector  LCX  =  half  the  space  DKZP)  and  in 
CX  take  C/x  =  PM  ;  then  the  line  B/x/x  is  the  involute  of 
BMA,  or  the  space  C/*/3  of  the  space  ADB. 

For  instance,  if  ADB  is  a  quadrant  of  a  circle,  the  line  ftnC 
is  a  semicircle. 

COR.  i.  It  is  to  be  observed  that  if  the  two  figures  ADB, 
ADG  are  analogous;  and  the  involutes  are  C/A/?,  Oy  ;  and 
if  C/x  :  Cv  =  DB  :  DG  ;  then,  reciprocally, 

$v  =  DG  :  DB. 


COR.  2.  The  converse  of  this  is  also  true. 


LECTURE  XII— APPENDIX  III      191 

COR.  3.  If  Cvy,  CS/3  are  analogous  suo  modo,  that  is  if, 
when  any  straight  line  CvS  is  drawn  through  C,  Cv  to  CS  is 
always  in  the  same  ratio ;  then  these  will  be  the  involutes 
of  similar  lines. 


PROBLEM  10 


Given  any  figure  /#C0,  bounded  by  the  straight  lines  G/3, 
C<£,  and  another  line  (3(j>  ;  it  is  required  to  draw  the  evolute. 

With  centre  C,  describe  any  circular  arc  LE  (making  with 
the  straight  lines  C/3,  Cc£  the  sector  LCE)  ;  then,  CK  being 
drawn  perpendicular  to  LC,  let  the  curve  /3YH  be  so  related 
to  the  straight  line  CK  that,  when  any  straight  line  C/x,Z  is 
drawn,  and  CO  is  taken  equal  to  the  arc  LZ,  and  OY  is  drawn 
perpendicular  to  CK,  OY  =  C/*.  Also,  let  the  curve  BMP 
be  so  related  to  the  straight  line  DA  that,  when  DP  is  equal 
to  space  C/3YO/CL,  and  PM  is  drawn  perpendicular  to  DA, 
then  PM  =  C/x  also.  Then  the  space  DBFA  is  the  evolute 
o 


Example.  —  Let  LZE  be  the  arc  of  a  circle  described  with 
centre  C,  and  /3/xC  a  spiral  of  such  a  kind  that,  if  the  straight 
line  C//.Z  is  drawn  in  any  manner,  the  arc  EZ  always  bears 
to  the  straight  line  C/x,  some  assigned  ratio  (say,  R  :  8).  It 
is  plain  that  the  line  /3YH  is  straight,  for  we  have  always 
EZ  (or  KO)  :  C/x  (or  OY)  =  R  :  8.  Hence,  the  evolute  BMP 
is  a  parabola,  since  the  parts  AP,  AD  of  the  axis  are  in  the 
same  ratio  as  the  spaces  KOY,  KG/?,  that  is,  as  the  squares 
on  OY,  C/?,  or  the  squares  on  PM,  DB. 


192  BARROW'S  GEOMETRICAL  LECTURES 

Corollaries 

Theorem  i.  If  on  the  figure  /3C<£  is  erected  a  cylinder 
having  its  altitude  equal  to  the  whole  circumference  of  the 
circle  whose  radius  is  CL;  then  the  cylinder  will  be  equal 
to  the  solid  produced  by  rotating  the  figure  C/3HK  about  CK. 

Theorem  2.  Let  AMB  be  any  curve  of  which  the  axis' is 
AD  and  the  base  is  DB,  and  AZL  a  curve  such  that,  when 
any  straight  line  ZPM  is  drawn,  PZ  =  V(2APM) ;  and  let 
OYY  be  another  curve  such  that,  when  the  straight  line 
ZPMY  is  produced  to  meet  it,  ZP2 :  R2  =  PM  :  PY.  Lastly, 
let  DL  :  R  =  R  :  LE,  and  through  E,  within  the  angle  LOG, 
describe  the  hyperbola  EXX  ;  let  the  straight  line  ZHX, 
drawn  parallel  to  AD,  meet  it  in  X.  Then  the  space  PDOY 
will  be  equal  to  the  hyperbolic  space  LXHE. 

Hence,  the  sum  of  all  such  as  PM/APM  =  2LEXH/R2. 

Theorem  3.  Let  AMB  be  any  curve  whose  axis  is  AD 
and  base  is  DB ;  and  let  the  curve  KZL  be  such  that,  if  any 
straight  line  R  is  taken,  and  an  arbitrary  line  ZPM  is  drawn 
parallel  to  BD,  ^APM  :  PM  =  R  :  PZ;  then  the  space  ADLK 
is  equal  to  the  rectangle  contained  by  R  and  2^/ADB,  or 

ADLK/2R  =  VADB.* 

Example. — Let  ADB  be  a  quadrant  of  a  circle;  then  the 
sum  of  all  such  as  PM/  ^/APM  =  J(2Dk  .  arc  AB). 

Theorem  4.  Let  AMB  be  any  curve  of  which  the  axis  is 
AD  and  the  base  is  DB,  and  let  EXK,  GYL  be  two  lines  so 

*  The  theorems  equivalent  to  Theorems  2  and  3  are  clear  enough,  even 
without  the  final  line  in  the  first  of  the  pair ;  Barro.w  intends  them  as 
standard  forms  in  integration. 


LECTURE  XII— APPENDIX  III       193 

related  that,  any  point  M  being  taken  in  the  curve,  and  the 
straight  lines  MPX,  MQY  being  drawn  respectively  parallel 
to  BD  and  AD,  and  it  being  supposed  that  MT  touches  the 
curve  AMB,  then  TP  :  PM  =  QY  :  PX.  Then  will  the  figures 
ADKE,  DBLG  be  equal  to  one  another. 


NOTE.  Of  all  the  propositions  so  far,  this  theorem  is  the 
most  fruitful  \  since  many  of  the  preceding  are  either  con- 
tained in  it  or  can  be  easily  deduced  from  it.  For,  suppose 
the  line  AMB  is  by  nature  indeterminate,  then  if  one  or 
other  of  the  curves  EXK,  GYL  is  determined  to  be  anything 
you  please,  there  will  result  from  the  supposition  some 
theorem  of  the  kind  of  which  we  have  given  a  considerable 
number  of  examples  already.  If,  for  instance,  the  line  GYL 
is  supposed  to  be  a  straight  line  making  with  BD  an  angle 
equal  to  half  a  right  angle  (in  which  case  the  points  D,  G 
are  taken  to  be  coincident),  then  we  get  the  theorem  of 
Lect.  XI,  §  i. 

If  GYL  is  a  line  parallel  to  DB,  we  have  Lect  XI,  §  u. 

Again,  if  PM  =  PX  (or  the  lines  AMB,  EKX  are  exactly 
the  same),  hence  follows  Lect.  XI,  §  10. 

Further  it  is  plain  from  the  theorem  that  for  any  given 
space  an  infinite  number  of  equal  spaces  of  a  different  kind 
can  be  easily  drawn  ;  thus,  if  the  space  DGLB  is  supposed  to 
be  a  quadrant  of  a  circle,  centre  D,  and  AMB  is  a  parabola 
whose  axis  is  AD,  we  get  this  property  of  the  curve  EKX  (by 
putting  DB  =  r,  AP  =  xt  PX  =  y,  and  k  for  the  semi-para- 
meter of  the  parabola  or  DB2/'2AB),  that  r^kl*  =  &x 

13 


194  BARROW'S  GEOMETRICAL  LECTURES 

If,  however,  AMB  is  supposed  to  be  a  hyperbola,  there 
will  be  produced  a  curve  EXK  of  another  kind. 

Moreover,  on  consideration,  I  blame  my  lack  of  fore- 
sight, in  that  I  did  not  give  this  theorem  in  the  first  place 
(it  and  those  that  follow,  of  which  the  reasoning  is  similar 
and  the  use  almost  equal);  and  then  from  it  (and  the  others 
that  are  added  directly  below),  as  I  see  can  be  done,  have 
deduced  the  whole  lot  of  the  others.  Nevertheless,  I  think 
that  this  sort  of  Phrygian  wisdom  is  not  unknown  either  to 
me  or  to  others  who  may  read  this  volume. 

NOTE 

When  I  first  considered  the  title-pages  of  the  volume 
from  which  I  have  made  the  translation,  I  was  struck  by 
the  fact  that  the  Lectiones  Opticce  had  directly  beneath  the 
main  title  the  words  "  Cantabrigicn  in  Scholis  publicis  habits  " 
(delivered  in  the  public  Schools  of  Cambridge),  whereas 
no  such  notification  appears  on  the  separate  title-page  of 
the  Lectiones  Geometries.  When  later  I  found  that  the 
title-page  of  the  Lectiones  Mathematics  also  bore  this  noti- 
fication, I  became  suspicious  that  at  any  rate  there  was  no 
direct  evidence  that  these  lectures  on  Geometry  had  ever 
been  delivered  as  professorial  lectures,  though  they  might 
have  been  given  to  his  students  by  Barrow  in  his  capacity  of 
college  fellow  and  lecturer.  As  I  considered  the  Preface, 
I  was  confirmed  in  this  opinion ;  and  the  above  note  would 
seem  to  corroborate  this  suggestion.  For  surely  if  these 
matters  had  been  given  in  University  Lectures  in  the  Schools, 
it  would  not  have  been  necessary  to  wait  till  they  were 
ready  for  press  before  Barrow  should  find  out  that  his 
Theorem  4  was  more  fruitful  and  general  than  all  the 
others.  His  own  words  contradict  the  supposition  that  he 
initially  did  not  know  this  theorem,  for  he  blames  his 
"want  of  foresight."  This  raises  the  point  as  to  the  exact 
date  when  Newton  was  shown  these  theorems;  this  has 
been  discussed  in  the  Introduction- 


LECTURE  XII— APPENDIX  III       195 

Theorem  5.  Let  ADB  be  any  space,  bounded  by  the 
straight  lines  DAE,  DQBK  and  the  curve  A  MB,  also  let  EXK, 
GYL  be  two  curves  so  related  that,  if  any  point  M  is  taken 
in  the  curve  AMB,  and  DMX  is  drawn,  and  DQ  =  DM,  and 
QYBL,  DG  are  drawn  perpendicular  to  DB,  and.DT  is  per- 
pendicular to  DM,  and  the  straight  line  MT  touches  the 
curve  AMB;  if,  I  say,  when  these  things  are  so,  TD  :  DM  = 
DM  .  QY  :  DX2;  then  shall  the  space  DGLB  be  double  the 
space  EDK. 

Theorem  6.  Again,  let  AMB  be  any  curve  of  which  the 
axis  is  AD  and  the  base  is  DB;  and  let  EXK,  HZO  be  two 
curves  so  related  to  one  another  and  the  axes  AD,  a/3  so 
related  to  one  another  that,  if  a  point  M  is  taken  anywhere 
on  the  curve  AMB,  and  MPX  is  drawn  perpendicular  to  AD, 
and  a//,  is  taken  equal  to  AM,  and  /xZ  is  drawn  perpendicular 
to  aft,  and  it  is  supposed  that  MT  touches  the  curve  AMB, 
and  cuts  DA  in  T,  then  TP  :  PM  =  /*Z  :  PX.  Then  the 
spaces  ADKE,  a/20H  shall  be  equal  to  one  another. 

Theorem  7.  Let  ADB  be  any  space,  bounded  by  the 
straight  lines  DAE,  DBK  and  the  curve  AMB;  also  let  EXK, 
HZO  be  two  curves  so  related  that,  if  any  point  M  is  taken  in 
the  curve  AMB,  and  the  straight  line  DMX  is  drawn,  and  a//, 
is  taken  equal  to  the  arc  AM,  and  /u,Z  is  drawn  perpendicular 
to  the  straight  line  a/3,  and  DT  is  perpendicular  to  DM, 
and  the  straight  line  MT  touches  the  curve  AMB,  then 
DT :  DM  =  DM  .  ftZ  :  DX2.  Then  shall  the  space  a/30H  be 
double  the  space  EDK. 

But  here  is  the  end  of  these  matters. 


LECTURE    XIII 

! 

[The  subject  of  this  lecture  is  a  discussion  of  the  roots 
of  certain  series  of  connected  equations.  These  are  very 
ingeniously  treated  and  are  exceedingly  interesting,  but 
have  no  bearing  on  the  matter  in  hand;  accordingly,  as 
my  space  is  limited,  I  have  omitted  them  altogether.] 

Laus  DEO  Optimo  Maximo 
FINIS 


In  the  second  edition,  published  in  1674,  .there  were 
added  the  three  problems  given  below,  together  with  a  set 
of  theorems  on  Maxima  and  Minima.  Problem  II  is  very 
interesting  on  account  of  the  difficulty  in  seeing  how  Barrow 
arrived  at  the  construction,  unless  he  did  so  algebraically. 

PROBLEM  I.  Let  any  line  AMB  be  given  (of  which  the 
axis  is  AD,  and  the  base  DB),  it  is  required  to  draw  a  curve 
ANE,  such  that  if  any  straight  line  MNG  is  drawn  parallel 
to  BD,  cutting  ANE  in  N,  then  the  curve  AN  shall  be  equal 
to  GM. 

The  curve  ANE  is  such  that  if  MT  touches  the  curve  AMB, 
and  NS  the  curve  ANE,  then  8G  :  GN  =  TG 


EXTRACTS  FROM  SECOND  EDITION    197 

PROBLEM  II.  With  the  rest  of  the  hypothesis  and  con- 
struction remaining  the  same,  let  now  the  curve  ANE  be 
required  to  be  such  that  the  arc  AN  shall  be  always  equal 
to  the  intercept  MN. 

Let  the  curve  ANE  be  such  that  SG  :  GN  =  2TG  .  GM  : 
(GM2  -TG2),  then  ANE  wjll  be  the  required  curve. 

PROBLEM  III.  Let  any  curve  DXX  be  given,  whose  axis 
is  DA;  it  is  required  to  find  a  curve  AMB  with  the  property 
that,  if  any  straight  line  GXM  is'drawn  perpendicular  to  AD, 
and  it  is  given  that  SMT  is  the  tangent  to  the  curve  AM, 
then  MS  =  GX. 

Clearly  the  ratio  TG  :  TM  (that  is,  the  ratio  of  GD  to  MS 
or  GX)  is  given ;  and  thus  the  ratio  TG  :  GM  is  also  given. 


Barrow  does  not  give  proofs  of  these  problems.  The 
only  geometrical  proof  of  the  second  I  can  make  out  is  as 
follows : — 

Draw  PQR  parallel  to  GNM,  cutting  the  curves  ANE,  AMB 
in  Q,  R  respectively,  and  draw  MW,  NV  parallel  to  AD  to 
meet  PQR  in  W,  R.  Then  we  have  NQ  -  RW  -  QV  from  the 
supposed  nature  of  the  curve  ;  also  from  the  several  differen- 
tial triangles,  we  have  RW/GP  -  MG/GT,  QV/GP  -  NG/GS, 
and  NQ/GP  =  NS/SP;  and  therefore 

NS.GT=  MG.GS-GT.NG. 

Squaring, 

(NG2  +  GS2).GT2  m  MG2.GS2-2MG.GS.GT.GN 
hence,  GS  .  (GM2  -  GT2)  =  2MG  .  GT  .  NG, 

or  GS:GN  =  2MG.GT:(GM2-GT2). 


But  I  can  hardly  imagine  Barrow  performing  the  opera- 
tion of  squaring,  unless  he  was  working  with  algebraic 
symbols ;  in  this  case  he  would  be  using  his  theorem  that 

(dsjdxf  =  i+(dyldx)*.      (Lect.  X,  §  5.) 


POSTSCRIPT 


Extracts  from  Standard  Authorities 

Since  this  volume  has  been  ready  for  press,  I  have  con- 
sulted the  following  authorities  for  verification  or  contra- 
diction of  my  suggestions  and  statements. 


ROUSE  BALL  (A  Short  Account  of  the  History 
of  Mathematics) 

(i).  "  It  seems  probable,  from  Newton's  remarks  in  the 
fluxional  controversy,  that  Newton's  additions  were  confined 
to  the  parts  "  (of  the  Lectiones  Optica  et  Geometriccz)  "  which 
dealt  with  the  Optics." 

(ii).  "The  lectures  for  1667  .  .  .  suggest  the  analysis 
by  which  Archimedes  was  led  to  his  chief  results." 

(iii).  "  Wallis,  in  a  tract  on  the  cycloid,  incidentally  gave 
the  rectification  of  the  semi-cubical  parabola  in  1659;  the 
problem  having  been  solved  by  Neil,  his  pupil,  in  1657; 
the  logarithmic  spiral  had  been  rectified  by  Torricelli "  (i.e. 
before  1647).  "The  next  curve  to  be  rectified  was  the 
cycloid;  this  was  done  by  Wren  in  1658." 

(This  contradicts  Williamson  entirely ;  I  suggest  that,  of 
the  two,  Ball  is  probably  the  more  correct,  if  only  for  the 
fact  that  this  would  explain  why  Barrow  did  not  remark  on 
the  fact  that  he  had  rectified  both  the  cycloid  and  the 
logarithmic  spiral.) 


POSTSCRIPT  199 

(iv).  The  only  thing  in  Barrow's  work  that  is  given  any 
special  notice  is  the  differential  triangle ;  since  Ball  states 
that  his  great  authority  for  the  time  antecedent  to  1758 
is  M.  Cantor's  monumental  work  Vorlesungen  iiber  die 
Geschichte  der  Mathematik,  it  would  appear  that  Cantor 
also  does  not  give  Barrow  the  credit  that  he  deserves. 

(v).  Fermat  had  the  approximation  to  the  binomial 
theorem ;  for  he  was  'able  to  state  that  the  limit  of 
e/{i  -  (i  -e)5/B},  when  e  is  evanescent,  is  3/5.  Since  we 
know  that  Fermat  had  occupied  himself  with  arithmetic 
and  geometric  means,  it  would  seem  probable  that  Barrow's 
equivalent  theorem  was  deduced  from  this  work  of  Fermat ; 
however,  Ball  states  that  these  theorems  of  Fermat  were  not 
published  until  after  his  death  in  1665,  whereas  Barrow's 
theorem  was,  I  have  endeavoured  to  show,  considerably 
anterior  to  this. 

(vi).  With  reference  to  the  Newton  controversy  we  have  : — 
"  It  is  said  by  those  who  question  Leibniz'  good  faith,  that 
to  a  man  of  his  ability  the  manuscript  (Newton's  De  Analyst 
per  Aequationes\  especially  if  supplemented  by  the  letter  of 
Dec.  10,  1672,  would  supply  sufficient  hints  to  give  him  a 
clue  to  the  methods  of  the  calculus,  though  as  the  fluxional 
notation  is  not  employed  in  it,  anyone  who  used  it  would 
have  to  invent  a  notation." 

(How  much  more  true  is  this  of  Barrow's  Lectures,  which 
contained  a  complete  set  of  standard  forms  and  rules,  and 
was  much  more  like  Leibniz'  method,  in  that  it  did  not  use 
series  but  gave  rules  that  would  work  through  substitutions  \ 
See  under  Gerhardt.) 

"  Essentially  it  is  Leibniz'  word  against  a  number  of 
suspicious  details  pointing  against  him." 

(I  hold  that  the  dates  are  almost  conclusive,  as  they  are 
given  in  the  fourth  paragraph  of  the  preface;  and  in  this 
I  do  not  by  any  means  suggest  that  Leibniz  lied,  as  will  be 
seen  under  Gerhardt.  A  mathematician,  having  Leibniz' 
object  and  point  of  view,  would  more  probably  consider 
that  Barrow's  work  and  influence  was  a  hindrance  rather 
than  a  help,  after  he  had  absorbed  the  fundamental  ideas.) 


200  BARROW'S  GEOMETRICAL  LECTURES 


Professor  LOVE  (Encyc.  Brit.  Xlth.  ed.,  Art. 
11  Infinitesimal  Calculus •.") 

(i).  "  Gregory  St  Vincent  was  the  first  to  show  the 
connection  between  the  area  under  the  hyperbola  and 
logarithms,  though  he  did  not  express  it  analytically. 
Mercator  used  the  connection  to  calculate  natural 
logarithms." 

(ii).  "Fermat,  to  differentiate  irrational  expressions,  first 
of  all  rationalized  them ;  and  although  in  other  works  he 
used  the  idea  of  substitution,  he  did  not  do  so  in  this  case." 

(iii).  "  The  Lectiones  Opticce  et  Geometriccc  were  apparently 
written  in  1663-4." 

(iv).  "Barrow  used  a  method  of  tangents  in  which  he 
compounded  two  velocities  in  the  direction  of  the  axes  of 
x  andjy  to  obtain  a  resultant  along  the  tangent  to  a  curve." 

"  In  an  appendix  to  this  book  he  gives  another  method 
which  differs  from  Fermat's  in  the  introduction  of  a  second 
differential." 

(Both  these  statements  are  rather  misleading.) 

(v).  "Newton  knew  to  start  with  in  1664  all  that  Barrow 
knew,  and  that  was  practically  all  that  was  known  about  the 
subject  at  that  time." 

(vi).  "  Leibniz  was  the  first  to  differentiate  a  logarithm 
and  an  exponential  in  1695." 
(Barrow  has  them  both  in  Lect.  XII,  App.  Ill,  Prob.  4.) 

(vii).  "  Roger  Cotes  was  the  first,  in  1722,  to  differentiate 
a  trigonometrical  function." 

(It  has  already  been  pointed  out  that  Barrow  explicitly 
differentiates  the  tangent,  and  the  figures  used  are  applic- 
able to  the  other  ratios ;  he  also  integrates  those  of  them 
which  are  not  thus  obtainable  by  his  inversion  theorem 
from  the  differentiations.  Also  in  one  case  he  integrates 
an  inverse  cosine,  though  he  hardly  sees  it  as  such.  With 
regard  to  the  date  1722,  as  Professor  Love  kindly  informed 


POSTSCRIPT  201 

me  on  my  writing  to  him,  this  is  the  date  of  the  posthumous 
publication  of  Cotes'  work ;  Professor  Love  referred  me  to 
the  passages  in  Cantor  from  which  the  information  was 
obtained.) 

(viii).  "The  integrating  curve  is  sometimes  referred  to 
as  the  Quadratrix." 

(This  is  Leibniz'  use  of  the  term,  and  not  Barrow's. 
With  Barrow,  the  Quadratrix  is  the  particular  curve  whose 
equation  is 

v  =  (r  -  x]  tan  Trx/zr.  ) 

There  are  a  host  of  other  things  both  in  agreement  with 
and  in  contradiction  of  my  statements  to  be  found  in  this 
erudite  article ;  nobody  who  is  at  all  interested  in  the 
subject  should  miss  reading  it.  But  I  have  only  room  for 
the  few  things  that  I  have  here  quoted. 


Dr  GERHARDT  (Editions  of  Le ibnizian  Manuscripts,  etc?) 

(i).  In  a  letter  to  the  Marquis  d'Hopital,  Leibniz  writes  : — 
"  I  recognize  that  Barrow  has  gone  very  far,  but  I  assure 
you,  Sir,  that  I  have  not  got  any  help  from  his  methods. 
As  I  have  recognized  publicly  those  things  for  which  I  am 
indebted  to  Huygens  and,  with  regard  to  infinite  series,  to 
Newton,  I  should  have  done  the  same  with  regard  to 
Barrow." 

(In  this  connection  it  is  to  be  remembered,  as  stated  in 
the  Preface,  that  Leibniz'  great  idea  of  the  calculus  was  the 
freeing  of  the  work  from  a  geometrical  figure  and  the  con- 
venient notation  of  his  calculus  of  differences.  Thus  he 
might  truly  have  received  no  help  from  Barrow  in  his 
estimation,  and  yet  might,  as  James  Bernoulli  stated  in 
the  Ada  Eruditorum  for  January  1691,  have  got  all  his 
fundamental  ideas  from  Barrow.  Later  Bernoulli  (Acta 
Eruditorum,  June  1691)  admitted  that  Leibniz  was  far  in 
advance  of  Barrow,  though  both  views  were  alike  in  some 
ways.) 

(ii).  Leibniz  (Historia  et  Origo  Calculi  Differ  en  tialis) 
states  that  he  obtained  his  "characteristic  triangle"  from 


202  BARROW'S  GEOMETRICAL  LECTURES 

some  work  of  Pascal  (alias  Dettonville),  and  not  from 
Barrow.  This  may  very  probably  be  the  case,  if  he  has 
not  given  a  wrong  date  for  his  reading  of  Barrow,  which 
he  states  to  have  been  1675;  this  would  not  seem  to  be 
an  altogether  unprecedented  proceeding  on  the  part  of 
Leibniz,  according  to  Cantor.  It  is  difficult  to  imagine 
that  Leibniz,  after  purchasing  a  copy  of  Barrow  on  the 
advice  of  Oldenburg,  especially  as  in  a  letter  to  Oldenburg 
of  April  1673  ne  mentions  the  fact  that  he  has  done  so, 
should  have  put  it  by  for  two  whole  years ;  unless  his 
geometrical  powers  were  not  at  the  time  equal  to  the  task 
of  finding  the  hidden  meaning  in  Barrow's  work. 

(iii).  Gerhardt  states  that  he  has  seen  the  copy  of  Barrow 
referred  to  in  the  Royal  Library  at  Hanover.  He  mentions 
the  fact  that  there  are  in  the  margins  notes  written  in 
Leibniz'  own  notation,  including  the  sign  of  integration. 
He  also  lays  stress  on  the  fact  that  opposite  the  Appendix  to 
Lect.  XI  there  are  the  Latin  words  for  "  knew  this  before." 
This  tells  against  Leibniz,  and  not  for  him,  for  this  Appendix 
refers  to  the  work  of  Hujgens,  which  of  course  Leibniz 
"  knew  before,"  and  Gerhardt  does  not  state  that  thcie 
words  occur  in  any  other  connection ;  hence  we  may  argue 
that  this  particular  section  was  the  only  one  that  Leibniz 
"  knew  before."  The  sign  of  integration,  though  I  cannot 
find  any  mention  of  it  before  1675,  means  nothing,  for  it 
might  be  added  on  a  second  reference,  after  Leibniz  had 
found  out  the  value  of  Barrow's  book.  A  striking  "coin- 
cidence" exists  in  the  fact  that  the  two  examples  that 
Leibniz  gives  of  the  application  of  his  calculus  to  geometry 
are  both  given  in  Barrow.  In  the  first,  the  figure  (on  the 
assumption  that  it  was  taken  from  Barrow)  has  been  altered 
in  every  conceivable  way ;  for  the  second,  a  theorem  of 
Gregory's  quoted  by  Barrow,  Leibniz  gives  no  figure,  and 
it  was  only  after  reference  to  B arrows  figure  that  I  could 
complete  Leibniz'  construction  from  the  verbal  directions 
he  gave.  This  looks  as  if  Leibniz  wrote  with  a  figure 
beside  him  that  was  already  drawn,  possibly  in  a  copy  of 
Gregory's  work,  or,  as  I  think,  from  Barrow's  figure.  I 
have  been  unable  to  ascertain  the  date  of  publication  of 
this  theorem  by  Gregory,  or  whether  there  was  any  chance 
of  its  getting  into  the  hands  of  Leibniz  in  the  original. 


POSTSCRIPT  203 


Professor  ZEUTHEN  (Geschichte  der  Mathematik  im 

XVI.  undXVILJahrhundert;  Deutsche  Ausgabe 

von  Raphael  Mayer). 

(i).  Oxford  and  Cambridge  seem  to  be  mixed  up  in  the 
historical  section,  for  it  is  stated  that  Barrow  was  Professor 
of  Greek  at  Oxford  and  Wallis  was  the  Professor  of 
Mathematics  at  Cambridge,  as  the  context  suggests  that  he 
was  Barrow's  tutor. 

(ii).  "...  he  produced  his  important  work,  the  Lectioms 
Mathematics,  a  continuation  of  the  Lectiones  Opticce;  this 
was  published,  with  the  assistance  of  Collins,  the  first 
edition  in  1669-70,  the  second  edition  in  1674." 

(Thus  Williamson's  error  is  repeated ;  it  would  be  inter- 
esting to  know  whether  Zeuthen  and  Williamson  obtained 
this  from  a  common  source,  and  also  what  that  source  was.) 

(iii).  "He"  (Leibniz)  "utilized  his  stay"  (in  London  in 
J67s)  "to  procure  the  Lectiones  of  Barrow,  which  Oldenburg 
had  brought  to  his  notice."  (See  under  Gerhardt.) 

(iv).  Zeuthen,  most  properly,  directs  far  more  attention 
to  the  inverse  nature  of  differentiation  and  integration,  as 
proved  by  Barrow,  than  to  the  differential  triangle.  But, 
by  his  repeated  reference  to  the  problem  of  Galileo,  he 
does  not  seem  to  have  perceived  the  fact  that  the  first  five 
lectures  were  added  as  supplementary  lectures.  Yet  he 
notes  the  fact  that  Barrow  does  not  adhere  to  the  kine- 
matical  idea  in  the  later  geometrical  constructions.  He 
also  calls  attention  to  the  generality  of  Barrow's  proofs. 

(v).  He  mentions  the  differentiation  of  a  quotient,  as 
given  in  the  integration  form  in  Lect.  XI,  but  appears  to 
have  missed  the  fact  that  the  rules  for  both  a  product  and 
a  quotient  have  been  given  implicitly  in  an  earlier  lecture. 

I  have  not  room  for  further  extracts  ;  each  reader  of  this 
volume  should  also  read  Zeuthen,  pp.  345-362,  if  he  has 
not  already  done  so.  What  he  finds  there  will  induce  him 
to  read  carefully  the  whole  of  this  excellent  history  of  the 
two  centuries  considered. 


204  BARROWS  GEOMETRICAL  LECTURES 


EDMUND  STONE  (Translation  of  Harrow's  Geometrical 
Lectures,  pub.  1735) 

This  translation  is  more  or  less  useless  for  my  purpose. 
First  of  all,  it  is  a  mere  translation,  without  commentary  of 
any  sort,  and  without  even  a  preface  by  Stone. 

The  title-page  given  states  that  the  translation  is  "from 
the  Latin  edition  revised,  corrected  and  amended  by  the 
late  Sir  Isaac  Newton."  If  this  refers  to  the  edition  of 
1670,  Stone  is  in  error.  But,  since  at  the  end  of  the  book, 
there  is  an  "  Addenda,"  in  which  are  given  several  theorems 
that  appeared  in  the  second  edition,  it  must  be  concluded 
that  Stone  used  the  1674  edition.  It  is  to  be  remarked 
that  these  theorems  are  on  maxima  and  minima,  and, 
according  to  the  set  given  by  Whewell,  only  form  a  part  of 
those  that  were  in  the  second  edition  of  Barrow  ;  some  two 
or  three  very  interesting  geometrical  theorems  being  omitted  ; 
one  of  these  is  extremely  hard  to  prove  by  Barrow's  methods, 
and  one  wonders  how  Barrow  got  his  theorem;  but  the  proof 
"drops  out"  by  the  use  of  dy/dx,  which  may  account  for 
Barrow  having  it,  but  not  for  Stone  omitting  it.  This  seems 
to  give  a  clue  as  well  to  an  altogether  unjustifiable  omission, 
by  either  Newton  or  Stone  (I  do  not  see  how  it  could  have 
been  Newton,  however)  at  the  end  of  the  Appendix  to 
Lect.  XI.  Two  theorems  have  been  omitted ;  their  in- 
clusion was  only  necessary  to  prove  a  third  and  final 
theorem  of  the  Appendix  as  it  stood  in  the  first  edition ; 
namely,  that  if  CED,  CFD  are  two  circular  segments  having 
a  common  chord  CGD,  and  an  axis  GFE,  then  the  ratio  of 
the  seg.  CED  to  the  seg.  CFD  is  greater  than  the  ratio  of  GE 
to  GF.  In  Stone  the  two  lemmas  are  omitted  and  the 
theorem  is  directly  contradicted.  The  proof  given  in  Stone 
depends  on  unsound  reasoning  equivalent  to  : — 

If«>£,     then     c  +  a:d+b>c\d, 

without  any  reference  to  the  value  of  the  ratio  of  c  to  d,  as 
compared  with  that  of  a  to  b.  Finally  the  theorem  as 
originally  given  is  correct,  as  can  be  verified  by  drawing 
and  measurement,  analytically,  or  geometrically. 

In  addition  to  this  alteration,  in  Stone  there  is  added  a 
passage  that  does  not  appear  in  the  first  edition,  nor  is  it 


POSTSCRIPT  205 

given  in  WhewelFs  edition.  "But  I  seem  to  hear  you 
crying  out  'aXXyv  8p9v  f^aXav^e,  Treat  of  something  else.'" 
In  a  table  of  errata  the  last  four  words  are  altered  to  "  Give 
us  something  else."  The  Greek  (there  should  be  no 
aspirate  on  the  first  word)  literally  means  "Shake  acorns 
from  another  oak."  If  this  alteration  was  made  by  Stone, 
the  addition  of  the  passage,  after  the  manner  of  Barrow,  is 
an  impertinence.  The  point  is  not,  however,  very  important 
in  itself,  but  taken  with  other  things,  points  out  the  com- 
parative uselessness  of  Stone's  translation  as  a  clue  to 
important  matters. 

The  whole  thing  seems  to  have  been  done  carelessly  and 
hastily ;  there  hardly  seems  to  have  been  any  attempt  to 
render  the  Latin  of  Barrow  into  the  best  contemporary 
English  ;  and  frequently  I  do  not  agree  with  Stone's  render- 
ing, a  remark  which  may  unfortunately  cut  either  way. 

Of  course  the  passage  may,  though  it  is  hardly  likely, 
have  been  added  by  Barrow ;  such  an  unimportant  state- 
ment would  hardly  have  been  added  in  those  days  of  dear 
books ;  it  is  also  to  be  noted  that  Whewell  does  not  give  it. 
The  point  could  only  be  settled  on  sight  of  the  edition  from 
which  Stone  made  his  translation.  Barrow,  however,  makes 
a  somewhat  similar  mistake  with  ratios  in  Lect.  IX,  §  10, 
and  Stone  passes  this  and  even  renders  it  wrongly.  This 
error  has  been  noted  on  page  107;  the  wrong  render- 
ing is  as  follows:  —  Barrow  has  FG  :  EF  +  TD  :  RD,  by 
which,  according  to  his  list  of  abbreviations,  he  means 
(FG/EF).  (TD/RD);  and  not,  as  Stone  renders  it,  FG/EF  + 
TD/RD,  without  noticing  that  this  does  not  make  sense  of 
the  proof. 

Perhaps  one  sample  of  the  carelessness  with  which  the 
book  has  been  revised  will  suffice :  he  has 

A  x  B  =  A  dividend  (sic)  by  B 

A 

=  A  multiplied  or  drawn  into  B ; 
B 

in  any  case  want  of  space  forbids  further  examples. 

It  is  this  untrustworthiness  that  make  it  impossible  to 
take  Stone's  statement  on  the  title-page  as  incontrovertible ; 
nor  another  statement  that  these  lectures  on  geometry  were 


206  BARROW'S  GEOMETRICAL  LECTURES 

delivered  as  Lucasian  Lectures ;  it  is  also  to  be  noted  that 
he  gives  as  Barrow's  Preface  the  one  already  referred  to  in 
the  Introduction  as  the  Preface  to  the  Optics  and  omits  the 
Preface  to  the  Geometry. 


WHEWELL  (The  Mathematical  Works  of  Isaac  Harrow, 
Camb.  Univ.  Press,  1860) 

(i).  Stress  is  only  laid  on  two  points  ;  one  of  course  is  the 
differential  triangle ;  the  other  is  the  "  mode  of  finding  the 
areas  of  curves  by  comparing  them  with  the  sum  of  the  in- 
scribed and  circumscribed  parallelograms,  leading  the  way 
to  Newton's  method  of  doing  the  same,  given  in  the  first 
section  of  the  Principia" 

(ii).  "  It  is  a  matter  of  difficulty  for  a  reader  in  these  days 
to  follow  out  the  complex  constructions  and  reasonings  of 
a  mathematician  of  Barrow's  time;  and  I  do  not  pretend 
that  I  have  in  all  cases  gone  through  them  to  my  satis- 
faction." (This  is  proof  positive  that  Whewell  did  not 
grasp  the  inner  meaning  of  Barrow's  work  ;  that  being  done, 
there  is,  I  think,  no  difficulty  at  all.) 

(iii).  The  title-page  of  the  Lectiones  Mathematics  states 
that  these  lectures  were  the  lectures  delivered  as  the 
Lucasian  Lectures  in  1664,  1665,  1666;  and  Lect.  XVI.  is 
headed 

MATHEMATICI    PROFESSORIS    LECTIONES 
(A.D.    MDCLXVI). 

(iv).  Lect.  XXIV  starts  the  work  on  the  method  of 
Archimedes,  which  would  thus  appear  to  be  the  lectures  for 
1667,  as  guessed  by  me,  and  as  stated  by  Ball. 

(v).  Whewell  gives  the  additions  that  appeared  in  the 
second  edition  of  1674.  These  consist  of  four  theorems, 
and  a  group  of  propositions  on  Maxima  and  Minima.  One 
theorem  is  noteworthy,  as  its  proof  depends  on  the  addition 
rule  for  differentiation  and  the  fact  that 


APPENDIX 

/.    Solution  of  a    Test  Question  on  Differentiation 
by  Barrow's  Method 

II.    Graphical  Integration  by  Barrow's  Method 
III.   Specimen  Pages  and  Plate 

I.  Test  Problem  suggested  by  Mr  Jourdain 

Given  any  four  functions  ,  represented  by  the  curves  <£</>,  60, 
££>  ££»  and  given  their  ordinates  -and  sub  tangents  for  any  one 
abscissa,  it  is  required  to  draw  the  tangent  for  this  abscissa 
to  the  curve  whose  ordinate  is  the  sum  (or  difference]  of  the 
square  root  of  the  product  of  the  ordinates  of  the  first  two 
curves  and  the  cube  root  of  the  quotient  of  the  ordinates  of 
the  other  two  curves. 

In  other  words,  differentiate 


The  figures  on  the  following  page  have  been  drawn  for 
y  =  ,J{sinx.loglQ(co5x)}±—  * 


(i).  Let  N0</>  be  the  ordinate  for  the  given  abscissa,  $F, 
01  the  given  subtangents  ;  let  TTTT  be  a  curve  such  that 
R.  NTT  =  N<£.  N0;  find  NP,  a  fourth  proportional  to 
NF+NT,  NF,  NT;  then  PTT  will  touch  the  curve  TTTT.  [See 
note  on  page  112,  rule  (i).] 


208  BARROW'S   GEOMETRICAL  LECTURES 


APPENDIX 


209 


(ii).  Let  N££  be  the  ordinate  for  the  given  abscissa,  £X, 
£Z  the  given  subtangents  ;  let  xx  be  a  curve  such  that 
Nx:R  =  N£:N£;  find  NQ,  a  fourth  proportional  to 
NZ-  NX,  NZ,  NX;  then  Qx  will  touch  the  curve  xx-  [See 
note  on  page  112,  rule  (ii).] 

(iii).  Let  pp  be  a  curve  whose  ordinate  varies  as  the  square 
root  of  the  ordinate  of  TTTT  ;  then  its  subtangent  NR  =  2NP 
(page  104). 

(iv).  Let  KK  be  a  curve  whose  ordinate  varies  as  the  cube 
root  of  the  ordinate  of  xx>  then  its  subtangent  NO  =  3NQ 
(page  104). 

(v).  Let  o-o-  be  a  curve  such  that  its  ordinate  is  the  sum 
of  the  ordinates  of  the  curves  KK,  pp  ;  take  N<r,  Nr  double  of 
NK,  Hp  respectively;  then  O,  Rr  are  the  tangents  to  the 
curves  whose  ordinates  are  double  those  of  the  curves  KK, 
pp  ;  let  these  tangents  meet  in  s  ;  then  so-  will  touch  the 
curve  crcr.  (See  note  on  page  100.) 

If  sd  is  drawn  perpendicular  to  RC  to  meet  it  in  d,  then 
d&  will  touch  the  curve  88,  whose  ordinate  is  the  difference 
between  the  ordinates  of  the  curves  KK,  pp.  (See  note  on 
page  100.) 


Geometrical 
Relations 


I     _    I    _  I 
NQ  "  NX~W. 

NR  =  2NP 
NC  =  3NQ 


Analytical  Equivalents 

if  «-*.»,  1.^1=1.^+*  ML 

u     ax      $     ax      v     ax 

Ifv=FK        -    —  =-      -*  --    & 
v  '  dx      %  '  dx       I  '  dx 

ITU--/*,         U/^lJ-=2«/^ 

I  dx  I  dx 


dx 


. 
dx 


Hence 


[ 


U_  du^  V_  _  dv 

MI  dx     'j    dx 


210  BARROWS  GEOMETRICAL  LECTURES 


APPENDIX  2H 

II.  The  Area  under  any  Curve 

(Lect.  XII,  App.  Ill,  Prob.  5) 

In  the  diagram  on  the  opposite  page,  the  curve  CFD  is 
a  given  curve,  or  a  curve  plotted  to  the  rectangular  axes 
BD,  BC,  which  Barrow  would  be  unable  to  integrate  by  any 
of  the  methods  he  has  given,  or,  in  fact,  could  give.  The 
curve  that  I  have  chosen  is  one  having  the  equation 
y  —  ^/(i  -x4),  and  the  problem  is  to  draw  a  curve  that 
shall  exhibit  graphically  the  integral  J  dxjy  for  all  values 
of  the  limits,  subject  to  the  condition  that  these  limits  must 
be  positive  numbers  and  not  greater  than  unity.  The  first 
step  is  to  construct  the  curve  GNE,  which  is  such  that 
r  =  ^(i  -  04),  for  which  the  method  of  construction  is 
obvious  from  the  diagram.  Comparing  this  with  the 
enunciation  of  Barrow's  Prob.  5,  the  curve  shown  in  the 
diagram  is  Barrow's  curve  turned  through  a  right  angle; 
thus,  the  point  N  is  also  the  point  T  in  Barrow's  enuncia- 
tion. Then  a  curve  has  to  be  constructed  such  that  the 
several  lines  DN  or  DT  are  the  respective  subtangents.  The 
curve  produced  is  BMA,  the  method  of  construction  being 
clearly  shown  in  the  figure ;  starting  with  B,  each  point  is 
successively  joined  to  its  corresponding  point  on  the  curve 
GNE  (so  that  MT  is  the  tangent  at  M)  and  to  the  next  point 
on  GNE,  and  the  point  midway  between  the  two  points  in 
which  these  cut  the  next  ray  from  D  is  taken  as  the  point 
on  the  curve  BMA. 

With  this  figure  the  area  represented  in  Leibniz'  nota- 
tion by 

f  i/V(i  -  x*}dx    or    J'i/v/(i  -  04K#  =  R/DM  -  R/DB ; 
for,  if  r  =/(0),    since  DN  =  r  =/(<9),  from  the  curve  GNE, 
and  DT  =  p2 .  dOjdp,  from  the  curve  BMA,  it  follows  that 

\dOlf (6)  =  jVp/p2  =  i/p,  for  all  limits. 
The  value   between   the  limits  o  and   i   works  out  as 
i/DA-i/DB,    which   is   found    from    the    diagram   to   be 
1/4-8    -i,  taking  DB  =  i,  that  is  1-304;   the  true  value 
is  (r(i/2).r(i/4)}/{4-r(3/4)}  =  1-31  about. 

It  is  only  suggested  that  this  was  the  purpose  of  Barrow's 
problem,  not  that  he  drew  such  a  figure  as  I  have  given. 


212  BARROWS  GEOMETRICAL  LECTURES 


III,  Specimen  Pages  and  Plate 

Two  specimen  pages  and  a  specimen  of  one  of  Barrow's 
plates  here  follow.  The  pages  show  the  signs  used  by 
Barrow  and  the  difficulty  introduced  by  inconvenient  or 
unusual  notation,  and  by  the  method  of  "running  on" 
the  argument  in  one  long  string,  with  interpolations. 
The  second  page  shows  Barrow's  algebraic  symbolism. 
Especially  note 

7  rm       i-  i, 

k.m  :  :  r .  —  =  EK 

k 

which  stands  for 

since    k-.m  =  r :  EK,     /.  EK  =^?. 

K 

The  specimen  plate  shows  the  quality  of  Barrow's  dia- 
grams. The  most  noticeable  figure  is  Fig.  176,  to  be 
considered  in  connection  with  Newton's  method  as  given 
in  the  Prindpia. 


APPENDIX  213 

L   B   C   T.      IX. 

D  E  F  concurrents  punftis  S,  T  .  erit  femper  D  T  =  z  D  S.  Quod 
fiPEfuntutcubiipfarum  DF,  erit  Temper  DT=  3  D  S;  ac  fi-  £ig-P?- 
mili  deiaceps  modo. 

X.  Sine  reftae  V  D,  T  B  concurrences  in  T,  quas  decufTet  pofiiio- 
nedatareaaDB;  tranfeant  etiam  per  B  line*  EBE,  FBF  tales,  FlS-  I0°- 
ut  ducla  quacunque  P  G  ad  D  B  parallcla,  fit  perpetuo  P  F  eodem  or- 
dine  media  Arithmetice  inter  PC,  P  E  ;  tangat  autem  B  R  curvam 
E  B  E  j  opprtet  lineae  FBF  tangentem  ad  B  determmare. 

Sumptis  N  M  (^ordinura  in  qaibus  font  P  F,  P  E  cxponentibus) 

fiat  N  x  T  D+         xRD.MxTD::RD.SD} 


tur  B  S  ;  hacc  curvam  FBF  contingec. 

Nam  utcunque  du£h  fit  P  G,  diftas  Hncas  Pecans  ut  vides.    JEftque 
EG.FG::(4JM.N.ergoFGxTJD.  EGxTD::NxTD. 
MxTD.    Item  EFxRD.EGxTD  ::  M  —  NxRD.  MX 
T  D.  Quaproptcr  (  antecedentes  conjungendo  )  erit  F  G  x  T  D  -»-  Wyfr:  Left 
EFxRD.EGxTD::NxTD-t-M-NxRD.MxTD. 
fhoc  eft)  '::^)RD.SD.(c;  Eft  aotemLGxTD-1-KLxRD.    .  .  _ 
KGxTD:;RD.Sv>D.    quare  FG  xTD-|-FF  x  RD  .  EG  x   CVA  %' 
TD::LGxTD-i-KLxRD.KGxTD.  hinc,  cum  fit  EG  00       vir. 
c-KG  jeritFGxTD  +  E  F  x  RDc-LG  x  TD-f-KL  x  RD-   OOtyf- 
velFG.EF  +  TD.RDcr-LG.KL-l-TD.RD.  feu(dem- 
pta  communi  ratione  )   F  G  .  E  F  c~  L  G  ,  K  L  .   vel  componendo 
EG.  EFc-KG.KL  (e)  c~EG.EL.  unde  eftEF^iEL.   (Or-  !««. 
itaque  punftum  L  extra  curvara  FBF  fitum  eft  j  adeoquc  liquet      V1I> 
Propofitum. 

XI.  Quinetiam,  rcliquis  ftantibus  iifdem,  fi  PF  fupponatur  ejuf- 
dem  ordinis  Geometrice  media  liquet  (plane  ficut  in  modo  przccdcn- 
ttbus)  eandem  B  S  curvam  FBF  contingere. 

Si  P  F  fit  e  fex  mcdiis  tertia,  feu  M  =  7  .  &  N  —  3  ; 


XII.  Patet  etiam,  accepto  qaolibet  in  curva  FBF  pun^lo  (c  eu  F  )    c  -a 
re&am  ad  hoc  tangentem  confimili  pafto  defignari.    -Nempe  per  F      '*"  ' 
ducatur  refta  P  G  ad  D  8  parallela  ,  fecans  curvam  E  B  E  ad  E  .  & 

per  E  ducatur  E  R  curvam  EBE  tangens;  fiatque  NxTP"'^  ^?  ?  x  RP. 

L  M  K 


214  BARROWS  GEOMETRICAL  LECTURES 

LECT.  x.  gj 

—  3  m  m  a  —  5  ffe  _  £  w  ,  ' 


3*'  = 


£xo»p.   IV. 

Sit  Quadratrix  C  M  V  (ad  circulum  C  E  B  pertinens  cui  centrum 
A  ,  )  cujus  axis  V  A  j  ordinal*  C  A  .  M  P  ad  V  A  perpendkula- 
res. 

Prfltraais  retfis  A  M  E,  A  N  F,  duclifque  rcftis  E  K,  F  L  ad  A  B 
perpcndicularibus  ,  dicantur  arcus  C  B  =  p  ,   radius  A  C  =  r  .  refta  Fig, 
AP  =  /  i  A  M  =  *  .  Eftquc  jam  C  A  arc.  C  B  :  :  N  R  .  arc.  F  E. 


&AM.MP::AE.EK;hoc 
eft,  *..»::  r         =  EK-  ircm  A  E  .  E  K  :  :  arc.  F  E  .L  K  .  hot 

AM.AE::AP.AK. 


hoc  eft  t.rs:f.^=AK.  ergo  ^/-A^AL. 

fabjeftis  tupcrfluis  )   -  A  L  <j  ,     adcoquc  LFf  ~ 
'^  fmpa. 


.. 

Eftautem  ACLq  .  QNq  :  :  ALq.LFq;  hoc  eft  Qj/—  r. 
Q;  w^4::ALq.LFq.  hoc  eft//—  2  f  ,.  w  ^  -t-  2  Wtf:  . 
rrf/;  —  zfmft.rrmm^zfmpa.  Unde  f  fublatis  ex  nor- 


raa  rcjcclancis  )     cmerget  ^ttAtlo^^-mm^—rrfA—rrme  .   feu 
vel  fubftituemlo  juxta 


kkf*—rrf*—rrmc  ;  vel  fubftituemlo  juxta  frafcrlftitm^kj^m  —  rrfnt 
-=.  rrmt  ;  vel  —  -  —  /  =  t  .       Hinc  colligitur  eflfe  reftam  A  T  ~ 


hoc  eft  (quoniana,  utnotum  eft,  A  V  =~     erit  A  T  — 
fcu,  AV.AM::AM.AT. 

M  2  Exewp. 


APPENDIX 


215 


INDEX 


PAGE 

Added  constant,  diff.  of  .  95 
Analogous  curves  .  .  81 
APOLLONIUS  .  i,  7,  n,  13, 

54,  57,  63 

Applied  lines       ...       43 
Arc,  of  circle       .         .         .      146 
approximations        .         .      147 
infinitesimal  =  tangent     .       61 
length,  see  Rectification. 
ARCHIMEDES       i,  7,  13,  54,  etc. 
ARISTOTLE         .        .     6,  n,  13 
Arithmetical    mean   greater 
than          geometrical 
mean         ...       85 
proportionals  .         .         -77 
Asymptotes          .         .         .       85 

BALL,  W.  W.  R.  .  .  198 
BARROW'S  mathematical 

works        ...         8 
symbols  .  .         .       22 

BERNOULLI  .  .  .201 
Bimedian  .  .  .  .152 
Binomial  approximation  .  87 
BRIGGS  .  .  .  183,  184 

CANTOR  .  .  .  .  199 
Cardioid  .  .  .  .100 
CAVALIERI  .  .  2 

Circular  functions,  diff.  of  .      123 
Spiral      .         .         .         .115 
Cissoid  of  Diocles        .        97 ,  109 
COLLINS     .         7,  14,  19,  26,  27 


Composite  motions 
Conchoid  of  Nicomedes 
Concurrent  motions 
Conical  surfaces  . 
COTES 


PAGE 

47 

95 

47 

173 

200 


Cycloid 
area  of    . 
rectification 


62,  198 

-     153 
161,  164,  177 


DESCARTES  3 

Descending  motion   or   de- 
scent 53 
DETTONVILLE    .        .        .     202 
D'HOPITAL       .        .        .201 
Difference  curve,  tangent  to     100 

diff.  of,  see  Laws. 
Differential  Triangle     13,  14,  120 
compared  with  fluxions     17,  1 8 
Differentiation    the    inverse 

of  integration  .  31,  117 
Directrix  ....  43 
Double  integral,  equivalent 

of     .         .         .         .133 

Equiangular  Spiral      .  139 

EUCLID      .        .        .13,  54,  57 
Exhaustions,  method  of       .170 
Exponent  of  a  mean  propor- 
tional       .         .          78,  83 
of  a  paraboliform     .          .      142 


FERMAT 
Fluxions 


4,  9, 


*99 

,  16 


INDEX 


217 


PAGE 

Fluxions,  proof  of  principle.     115 

Fractional  indices        .         .       1 1 

powers,  diff.  of        .         .104 

integ.  of     .         .128 

GALILEO  .  i,  4,  13,  58,  203 
Generation  of  magnitudes  .  35 
Genetrix  ....  43 
Geometrical  proportionals  .  77 
GERHARDT  .  .  .  201 
Graphical  integration  .  .  32 
GREGORY,  James  (of  Aber- 
deen) .  .  13,  131 
involutes  and  evolutes  .  190 
GULDIN  ....  2 

HUYGENS    .  .  13,  141,  201 

Hyperbola,     approximation 

to  curve    ...       68 
determination       of       an 

asymptote         .          69,  73 

Index  notation  .  .  .  3,  1 1 
of  a  mean  proportional  78,  83 
Indivisibles  ...  2 
Infinite  velocity,  case  of  .  59 
Integration,  method  of 

Cavalieri  .         .         .125 
inverse  of  differentiation  31,  135 


JESSOP 

KAUFMANN 
KEPLER 


186 

i 


Laws  for  differentiation   of 
a  product,   quotient, 
and  sum  ...       31 
LEIBNIZ      .        '5*95  200,  202 
Logarithmic  differentiation  .      106 
Spiral      .         .         .      139,  198 


PAGE 

LOVE  ....  200 
Lucasian  Lectures  6,  7,  194,  206 

Maximum  and  minimum       2,  32, 

63,  149 

Mean  proportionals  .  .  77 
MERCATOR  .  .  .  186 
METIUS'  ratio  for  IT  150,  151,  154 

NEIL.  .  .  .  138,  198 
NEWTON  3,  9,  16-20,  26,  199,  200 
Normals  or  perpendiculars  .  63 


Order  of  mean  proportional 

OUGHTRED 
OVERTON    . 


77 
5 

20 
154 


TT,  limits  for          .       150,  151, 
Paraboliforms,      centre      of 

gravity      .         .         .142 

tangent  construction  14,  104 
PASCAL  ...  2,  202 
Polar  subtangent  .  .  1 1 1 
Power,  differentiation  of  .  104 
Preface  to  the  Geometry  .  27 

to  the  Optics  ...  25 
Product  curve,  tangent  to  .  112 

diff.  of,  see  Laws. 
Properties      of     continuous 

curves        .         .         60-65 

Quadratrix.         48,  118,  201,  214 
Quadrature    of    the    hyper- 
bola         .         .      1 80,  1 86 

theorems  depending  on  165,  185 
Quotient  curve,  tangent  to  .  112 

diff.  of,  see  Laws. 

Reciprocal,  diff.  of  -94 

Rectification,      fundamental 

theorem    .         .        32,  115 
general  theorems      .         .      155 


218  BARROW'S  GEOMETRICAL  LECTURES 


RICCI 

ROBERVAL 
Root,  diff.  of      . 
Rotation,  mode  of  motion 


PAGE 

149 

2 

104 

42 


ST  VINCENT      .    n,  13,  72,  200 
Secants,    integration    theor- 
ems         .         .      165,  167 
Second    edition,    additional 

theorems .         .         .196 
Segment  of  circle  and  hyper- 
bola         .         .         .      146 
Semi-cubical  parabola       162,  198 
Spiral  of  Archimedes  48,  115,  119 
Standard  forms,  see  pp.  30-32. 
STONE        ....     204 
Subtangent          .         .         .106 
Sum  curve,  tangent  to          .100 

diff.  of,  see  Laws. 
Symbols,  Barrow's  list  of     .       22 


TACQUET   . 


PAGE 

Tangency,  criterion  of         .  90 

Tangents,  definitions  of       .  3 

integration  theorems  on  .  166 

THEODOSIUS  ...  7 
Time,  see  Lecture  I. 

TORRICELLI'S  Problem  .  58 
Translation,  a  mode  of 

motion      ...  42 
Trigonometrical  approxima- 
tions        ...  32 

ratios,  diff.  of  .         .         .122 

Trimedian  .         .         .  152 

VAN  HURAET  .  .  .162 
Velocity,  laws  of  .  .40 
WALLIS  2,  n,  13,  138,  162,  198 
WHEWELL  .  .  .  206 
WILLIAMSON  ...  6 
WREN  .  .  139,  179,  198 


4      ZEUTHEN  . 


203 


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